Refactoring calculation caches

Suppose I have a class Group, like so:

class Group {
   ...
   mutable std::map<Input, Output> complexCalculationCache;
public:
   ...

   bool operator==(const Group& rhs); // excludes
complexCalculationCache in comparison since it's irrelevant

   Output complexCalculation(const Input& input); // does the
complex calculation if it's not in the cache, uses the cache otherwise
};

What I would like to do is to separate out the
complexCalculationCache, since it's really only used in
Group::complexCalculation() and nowhere else, and is ultimately
hindering my efforts to make compositions involving Group (ie. keeping
G1's cache and G2's caches in sync if G1 == G2). Firstly, is this a
good idea? Secondly, if it is, then I'd probably need to have a
singleton cache manager of sorts (perhaps something backed by
std::unordered_map<Group, std::map<Input, Output>> with a hash
function for Group). Can someone point me to the best way to do this?

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