Equivariant Diffeomorphism $S^2 times S^3$ to itself with respect to the following $mathbb{Z}_4$ action

Here's an example that makes use of the fact that the unit quaterions act by rotations on $S^2$.

Consider $S^2times S^3$ as a subset of $mathbb{H}_{im}timesmathbb{H}$, where $mathbb{H}_{im}$ denotes the subspace of purely imaginary quaterions. Note that the unit quaternions act on $mathbb{H}_{im}$ by conjugation $alphacdotbeta=alphabetaalpha^*$, which have the effect of Euclidean rotations. We can choose the identification $mathbb{R}^3congmathbb{H}_{im}$ so that the matrix $R$ is given by the action of $j$ (since unit imaginary quaterions act by $180^circ$ rotations). Define a map $f:mathbb{H}_{im}timesmathbb{H}tomathbb{H}_{im}timesmathbb{H}$ by $$ f(alpha,beta)=(betaalphabeta^*,beta) $$ This map can be smoothy restricted to $S^2times S^3subsetmathbb{H}_{im}timesmathbb{H}$, and has an inverse given by $f^{-1}(alpha,beta)=(beta^*alphabeta,beta)$. Equivariance is straightforward to verify.