My question is essentially in group theory but came from working on a problem in Polya Theory. Let $G$ be a finite group. We know then that $G$ is isomorphic to some subgroup of $H \subseteq S_{k}$ for some $k \in \mathbb{N}$. This $H$ may not be unique, so let $H' \subseteq S_{k'}$ be an isomorphic subgroup of $S_{k'}$ not necessarily equal to $H$ or with $k = k'$. Let $\phi: G \rightarrow H$ denote the isomorphism between $G$ and $H$. It is well known that any permutation has a cycle decomposition. Define $c_{k}: S_{k} \rightarrow \mathbb{N}$ such that $c(\pi)$ is the number of cycles in the decomposition of $\pi$. Based on a result from Polya Theory, I expect that $$c_{k}(\phi(g)) = c_{k'}(\phi'(g))$$ for all $g \in G$.
More precisely given any element of a group, is there a well-defined function representing the number of cycles in its decomposition in any subgroup of a symmetric group that it maps to under isomorphism? If so, why?
No. To take a silly counterexample, take $g$ to be the identity, then $c_k(g)=k$, so the value depends on $k$. An only slightly less silly example, take $G$ to be cyclic of order $2$, and $g$ its generator. Then the only restriction is that $\phi(g)$ should be an element of order $2$ in $S_k$, so all its cycles have length $2$ or $1$, but that tells you nothing about the number of cycles. (Even for fixed $k$, you could have a transposition, a double transposition, etc.)