I'm looking for simple examples of calculations of equivariant homology and of equivariant bordism.

I have a finite group G acting on an CW-complex X. I would like to calculate the equivariant homology $H_\ast^G(X)= H_\ast(X\times_G EG)$. In all of the examples I know, either the calculation is degenerate, for example because action of $G$ is taken to be free in which case $H_\ast^G(X)\simeq H_\ast(X/G)$, or else the setting is quite complex (string topology for instance) and I have trouble following. What I need is a good "second example".

I am looking first for a useful example of a simple non-degenerate equivariant homology calculation. One example I am specifically interested in is to calculate $H_\ast^{C_p}(C_q)$, where $C_p\ltimes C_q$ is a metacyclic group, so that $C_p$ is a cyclic group of order p acting on $C_q$ (with addition as the group operation) by multiplication by a $p$th root of unity in $C_q$. For example, $C_2$ would act on $C_7$ by multiplication by $6$, and $C_3$ would act on $C_7$ by multiplication by $2$. The action is not free (it fixes $0\in C_q$) so this example is not degenerate, yet the groups are small and finite.

I have the same question with regard to equivariant bordism. In the above example for instance, consider smooth maps from CW-complexes to the Eilenberg-Maclane space $K(C_q,1)$, where two such maps are considered equivalent if their image differs by a smooth $C_p$-action on $K(C_q,1)$. How to calculate $\Omega_\ast^{C_p}(C_q)$?