Assume a finite group $G$ acts on a topological space $X$ and $A\subseteq X$. Denote by $q$ the quotient map from $X$ to the orbit space $X/G$ (we take the quotient topology). Moreover, let $H_n(A)=0$ for some $n\geq 1$.
Let us consider a subset $q(A)$ of $X/G$ with the induced subset topology.
When is it the case that $H_n(q(A))=0$? (I mean especially group properties of $G$) If the answer is not possible in general, it would be nice to point out specific situations too (e.g. $X$ - manifold, smooth manifold, Lie group, etc.)
The analogous problem for cohomology is also interesting to me.