Prove that if $G$ is finite, then any neighborhood of a $G$-invariant subset of $X$ contains a $G$- invariant neighborhood

Question

Let $G$ be a group acting on a topological space $X$. Prove that if $G$ is finite, then any neighborhood of a
$G$-invariant subset of $X$ contains a $G$-invariant neighborhood.

I have no idea even how to start. Can anyone please help me by giving a hint?

Answer 1

Let $U$ be an open neighborhood of the $G$-invariant subset $A$. Now define $W=\bigcap_{g\in G}gU$. Can you show that this set is a $G$-invariant neighborhood of $A$?