Given two group actions $\alpha:G\to\mathrm{Perm}(X)$ and $\beta:G\to\mathrm{Perm}(Y)$, say the $G$-sets $X$ and $Y$ are cycle-similar (let's denote $X\sim Y$, since it's an equivalence relation) if $\alpha(g)$ and $\beta(g)$ are the same cycle type for each group element $g\in G$. If $X\cong Y$ are isomorphic as $G$-sets then $X\sim Y$ are cycle-similar but it ought to be possible for $X\sim Y$ even while $X\not\cong Y$.
(It's noted here that $\mathrm{Out}(G)$ can have inequivalent actions on the sets $\mathrm{Class}(G)$ of conjugacy classes of $G$ and $\mathrm{Irr}(G)$ of irreducible representations of $G$, but the character table is invariant so Brauer's permutation lemma shows every $\alpha\in\mathrm{Out}(G)$ has the same cycle type on both sets.)
I am interested in examples where $X\not\cong Y$ but $X\sim Y$. Is there a way to produce examples of such pairs? Are there small, understandable examples? Can one classify "minimal" examples?
By Burnside's lemma, $|X/G|=|Y/G|$, so $X$ and $Y$ have the same number of orbits, and wlog they share no isomorphic orbits. Also, their cycle index polynomials $Z_{\alpha(G)}(t)$ and $Z_{\beta(G)}(t)$ must be the same, but in principle one could have $Z_{\alpha(G)}(t)=Z_{\beta(G)}(t)$ while $X\not\sim Y$, but finding examples of that is likely even more difficult.
If $G/H\sim G/K$ (so, one orbit each), then each $g$ must have the same index with respect to every conjugate $aHa^{-1}$. IOW, for $aH,gaH,\cdots,g^{k-1}aH$ to be a cycle, $k$ must be the smallest exponent for which $g^k\in aHa^{-1}$. Not sure how to turn this into an example.
Also, $X\sim Y$ implies $Z\times X\sim Z\times Y$ so the set $\{[X]-[Y]\mid X\sim Y\}$ of virtual differences ought to be an ideal in the Burnside ring $\mathrm{Burn}(G)$ of $G$. Does this ideal have a name?