Note that $ \langle a,b|a^2b^{-1}a^{-2} b \rangle =\pi_1 ( X):=G$ where $X=\mathbb{R}P^2\sharp \mathbb{R}P^2$.
So $a^2,\ b$ commute in $G$ so that $G$ contains $H:=\mathbb{Z}^2$.
In further note that $[H:G]=2$. That is $G$ may be a semi-direct between $\mathbb{Z}^2$ and $\mathbb{Z}_2$ or a free product. $H$ is a normal subgroup ? Thank you in advace.