Fix Style/ClassEqualityComparison autocorrection within module

changelog/fix_fix_styleclassequalitycomparison.md

@@ -0,0 +1 @@
+* [#10850](https://github.com/rubocop/rubocop/pull/10850): Fix `Style/ClassEqualityComparison` autocorrection within module. ([@r7kamura][])

lib/rubocop/cop/style/class_equality_comparison.rb

@@ -97,7 +97,9 @@ def class_name(class_node, node)
           if node.children.first.method?(:name)
             return class_node.receiver.source if class_node.receiver
 
-            class_node.source.delete('"').delete("'")
+            value = class_node.source.delete('"').delete("'")
+            value.prepend('::') if class_node.each_ancestor(:class, :module).any?
+            value
           else
             class_node.source
           end

spec/rubocop/cop/style/class_equality_comparison_spec.rb

@@ -100,4 +100,31 @@ def equal?(other)
       RUBY
     end
   end
+
+  context 'with String comparison in module' do
+    it 'registers and corrects an offense' do
+      expect_offense(<<~RUBY)
+        module Foo
+          def bar?(value)
+            bar.class.name == 'Bar'
+                ^^^^^^^^^^^^^^^^^^^ Use `instance_of?(::Bar)` instead of comparing classes.
+          end
+
+          class Bar
+          end
+        end
+      RUBY
+
+      expect_correction(<<~RUBY)
+        module Foo
+          def bar?(value)
+            bar.instance_of?(::Bar)
+          end
+
+          class Bar
+          end
+        end
+      RUBY
+    end
+  end
 end