diff --git a/lib/checkclass.cpp b/lib/checkclass.cpp
index 082a36b56..53e6cb2fd 100644
--- a/lib/checkclass.cpp
+++ b/lib/checkclass.cpp
@@ -2306,6 +2306,19 @@ bool CheckClass::checkConstFunc(const Scope *scope, const Function *func, bool&
                 return false;
             memberAccessed = true;
         }
+        bool mayModifyArgs = true;
+        if (const Function* f = funcTok->function()) { // TODO: improve (we bail out if there is any possible modification of any argument)
+            const std::vector<const Token*> args = getArguments(funcTok);
+            const auto argMax = std::min<nonneg int>(args.size(), f->argCount());
+            mayModifyArgs = false;
+            for (nonneg int argIndex = 0; argIndex < argMax; ++argIndex) {
+                const Variable* const argVar = f->getArgumentVar(argIndex);
+                if (!argVar || !argVar->valueType() || ((argVar->valueType()->pointer || argVar->valueType()->reference != Reference::None) && !argVar->isConst())) {
+                    mayModifyArgs = true;
+                    break;
+                }
+            }
+        }
         // Member variable given as parameter
         const Token *lpar = funcTok->next();
         if (Token::simpleMatch(lpar, "( ) ("))
@@ -2313,9 +2326,9 @@ bool CheckClass::checkConstFunc(const Scope *scope, const Function *func, bool&
         for (const Token* tok = lpar->next(); tok && tok != funcTok->next()->link(); tok = tok->next()) {
             if (tok->str() == "(")
                 tok = tok->link();
-            else if ((tok->isName() && isMemberVar(scope, tok)) || (tok->isUnaryOp("&") && (tok = tok->astOperand1()))) {
+            else if ((tok->isName() && isMemberVar(scope, tok)) || (tok->isUnaryOp("&") && (tok = tok->astOperand1()) && isMemberVar(scope, tok))) {
                 const Variable* var = tok->variable();
-                if (!var || !var->isMutable())
+                if ((!var || !var->isMutable()) && mayModifyArgs)
                     return false; // TODO: Only bailout if function takes argument as non-const reference
             }
         }
diff --git a/lib/cppcheck.h b/lib/cppcheck.h
index e161f94ae..07469b8ae 100644
--- a/lib/cppcheck.h
+++ b/lib/cppcheck.h
@@ -145,7 +145,7 @@ public:
     bool isUnusedFunctionCheckEnabled() const;
 
     /** Remove *.ctu-info files */
-    void removeCtuInfoFiles(const std::map<std::string, std::size_t>& files);
+    void removeCtuInfoFiles(const std::map<std::string, std::size_t>& files); // cppcheck-suppress functionConst // has side effects
 
 private:
     /** Are there "simple" rules */
diff --git a/lib/symboldatabase.h b/lib/symboldatabase.h
index a28a44efc..c6d678092 100644
--- a/lib/symboldatabase.h
+++ b/lib/symboldatabase.h
@@ -1474,12 +1474,12 @@ private:
     void createSymbolDatabaseNeedInitialization();
     void createSymbolDatabaseVariableSymbolTable();
     void createSymbolDatabaseSetScopePointers();
-    void createSymbolDatabaseSetFunctionPointers(bool firstPass);
+    void createSymbolDatabaseSetFunctionPointers(bool firstPass); // cppcheck-suppress functionConst // has side effects
     void createSymbolDatabaseSetVariablePointers();
     // cppcheck-suppress functionConst
     void createSymbolDatabaseSetTypePointers();
     void createSymbolDatabaseSetSmartPointerType();
-    void createSymbolDatabaseEnums();
+    void createSymbolDatabaseEnums(); // cppcheck-suppress functionConst // has side effects
     void createSymbolDatabaseEscapeFunctions();
     // cppcheck-suppress functionConst
     void createSymbolDatabaseIncompleteVars();
diff --git a/lib/tokenize.h b/lib/tokenize.h
index 0ba1f9705..3edd17f17 100644
--- a/lib/tokenize.h
+++ b/lib/tokenize.h
@@ -586,7 +586,7 @@ private:
 
     void unsupportedTypedef(const Token *tok) const;
 
-    void setVarIdClassDeclaration(const Token * const startToken,
+    void setVarIdClassDeclaration(const Token * const startToken, // cppcheck-suppress functionConst // has side effects
                                   VariableMap &variableMap,
                                   const nonneg int scopeStartVarId,
                                   std::map<nonneg int, std::map<std::string, nonneg int>>& structMembers);
@@ -595,7 +595,7 @@ private:
                                std::map<nonneg int, std::map<std::string, nonneg int>>& structMembers,
                                nonneg int &varId) const;
 
-    void setVarIdClassFunction(const std::string &classname,
+    void setVarIdClassFunction(const std::string &classname, // cppcheck-suppress functionConst // has side effects
                                Token * const startToken,
                                const Token * const endToken,
                                const std::map<std::string, nonneg int> &varlist,
diff --git a/test/testclass.cpp b/test/testclass.cpp
index cc784f958..7d6ce802c 100644
--- a/test/testclass.cpp
+++ b/test/testclass.cpp
@@ -195,7 +195,9 @@ private:
         TEST_CASE(const79); // ticket #9861
         TEST_CASE(const80); // ticket #11328
         TEST_CASE(const81); // ticket #11330
-        TEST_CASE(const82);
+        TEST_CASE(const82); // ticket #11513
+        TEST_CASE(const83);
+
         TEST_CASE(const_handleDefaultParameters);
         TEST_CASE(const_passThisToMemberOfOtherClass);
         TEST_CASE(assigningPointerToPointerIsNotAConstOperation);
@@ -6322,7 +6324,26 @@ private:
         ASSERT_EQUALS("", errout.str());
     }
 
-    void const82() {
+    void const82() { // #11513
+        checkConst("struct S {\n"
+                   "    int i;\n"
+                   "    void h(bool) const;\n"
+                   "    void g() { h(i == 1); }\n"
+                   "};\n");
+        ASSERT_EQUALS("[test.cpp:4]: (style, inconclusive) Technically the member function 'S::g' can be const.\n",
+                      errout.str());
+
+        checkConst("struct S {\n"
+                   "    int i;\n"
+                   "    void h(int, int*) const;\n"
+                   "    void g() { int a; h(i, &a); }\n"
+                   "};\n");
+        TODO_ASSERT_EQUALS("[test.cpp:4]: (style, inconclusive) Technically the member function 'S::g' can be const.\n",
+                           "",
+                           errout.str());
+    }
+
+    void const83() {
         checkConst("struct S {\n"
                    "    int i1, i2;\n"
                    "    void f(bool b);\n"
diff --git a/test/testsuppressions.cpp b/test/testsuppressions.cpp
index 416108fdd..4c77752e4 100644
--- a/test/testsuppressions.cpp
+++ b/test/testsuppressions.cpp
@@ -561,7 +561,7 @@ private:
         ASSERT_EQUALS(true, suppressions6.isSuppressed(errorMessage("abc", "test.cpp", 123)));
     }
 
-    void inlinesuppress() {
+    void inlinesuppress() const {
         Suppressions::Suppression s;
         std::string msg;
         ASSERT_EQUALS(false, s.parseComment("/* some text */", &msg));
@@ -593,7 +593,7 @@ private:
         ASSERT_EQUALS("[test.cpp:4]: (error) Uninitialized variable: a\n", errout.str());
     }
 
-    void inlinesuppress_comment() {
+    void inlinesuppress_comment() const {
         Suppressions::Suppression s;
         std::string errMsg;
         ASSERT_EQUALS(true, s.parseComment("// cppcheck-suppress abc ; some comment", &errMsg));
@@ -604,7 +604,7 @@ private:
         ASSERT_EQUALS("", errMsg);
     }
 
-    void multi_inlinesuppress() {
+    void multi_inlinesuppress() const {
         std::vector<Suppressions::Suppression> suppressions;
         std::string errMsg;
 
@@ -669,7 +669,7 @@ private:
         ASSERT_EQUALS(false, errMsg.empty());
     }
 
-    void multi_inlinesuppress_comment() {
+    void multi_inlinesuppress_comment() const {
         std::vector<Suppressions::Suppression> suppressions;
         std::string errMsg;
 
diff --git a/test/testsymboldatabase.cpp b/test/testsymboldatabase.cpp
index 0d4990bf3..959977bef 100644
--- a/test/testsymboldatabase.cpp
+++ b/test/testsymboldatabase.cpp
@@ -7171,7 +7171,7 @@ private:
         ASSERT(tok->tokAt(2)->function());
     }
 
-    void valueTypeMatchParameter() {
+    void valueTypeMatchParameter() const {
         ValueType vt_int(ValueType::Sign::SIGNED, ValueType::Type::INT, 0);
         ValueType vt_const_int(ValueType::Sign::SIGNED, ValueType::Type::INT, 0, 1);
         ASSERT_EQUALS((int)ValueType::MatchResult::SAME, (int)ValueType::matchParameter(&vt_int, &vt_int));