Let $G$ be a group and $H$ be a subgroup of $G$ such that for any $x\in G$ we have $x^2\in H$. prove that $H$ is normal in $G$.

I think this true, but can not prove it. for example this is true for $$D\_8 = <a,b \space | a^4=b^2=1, \space ab = ba^{-1}>$$