$\forall x \in G, \ \forall h \in H: \ xhx^{-1}=h^{-1}(hx)^2(x^{-1})^2 \in H. \ \ \ \square$
since $Hg=Hg^{-1}, \ \forall g \in G,$ for any $x,y \in G$ we'll have: $HxHy=Hxy=H(xy)^{-1}=Hy^{-1}x^{-1}=Hy^{-1}Hx^{-1}=HyHx. \ \ \square$