How can I decide whether two groups defined by finite presentations are (or not) isomorphic?

Question

I have the groups $G_1,G_2$ with presentations $$G_1 = \langle x,y : (y^2x)^2 = x^2, (x^2 y )^2 = y^{-2} \rangle = \langle x,y : x^{-1}y^2 x = y^{-2}, yx^2y^3 = x^{-2} \rangle \\ G_2 = \langle x,y : (y^2x)^2 = x^2, (x^2 y )^2 = y^{2} \rangle = \langle x,y : xy^2x^{-1} = y^{-2}, y^{-1}x^2y = x^{-2} \rangle.$$ The both groups are free amalgamated products of two klein bottle groups over $\mathbb{Z}^2$ of the following way. Taking the presentations $$H_1 = \langle r,x : r^2 = x^2 \rangle, \\ H_2 = \langle s,y : s^2 = y^{-2} \rangle,$$ it follows that $ \langle rx^{-1}, x^2 \rangle\cong \langle sy^{-1} , y^2 \rangle \cong \mathbb{Z}^2$, and with the isomorphism $\phi : \langle rx^{-1}, x^2 \rangle\to\langle sy^{-1} , y^2 \rangle$ as $\phi(rx^{-1}) = y^2$ and $\phi (x^2) = sy^{-1}$, then $G_1 = H_1 \ast_{\mathbb{Z}^2} H_2$. The case of $G_2$ is all the same, except for $y^2$ instead of $y^{-2}$ in the presentation of $H_2$. My question is, How can I show that $G_1$ and $G_2$ are isomorphic or not?