If $H \le G$ and $H$ has index 2 in $G$ then $a^2 \in H, \forall a\in G$

Question

Claim: If $H \le G$ and $H$ has index 2 in $G$ then $a^2 \in H, \forall a\in G$

My attempt: There will be exactly two cosets lets call $H$ and $aH$ respectively. If $a\in H$ then $a^2\in H$ because of closure property of $H$. Second case when $a\in aH$ then $aHaH = a^2H$ so now how to prove that $a^2 \in H$

Please Help.

Answer 1

Hint: If $a^2H = aH$, then $aH = H$.