Suppose the topological groups $h, G$ satisfy that $H < G$ and that $[G \colon H] < \infty$. Let $V$ be a $K$-vector space on which $H$ acts continuously. Then we consider the induced $G$-representation $W \colon= {\Bbb Z}[G] \otimes_{{\Bbb Z}[H]} V$.
Q. How can I explicitly understand the equality $W^{H} = \underset{[G \colon H]}{\oplus} V^{H}$?
What if $[G \colon H] = 2$?