Consider the covering space of $S^1 \vee S^1$ in $(1)$. Then distinct loops in $(1)$ are represented by $\langle a, b^2, bab^{-1} \rangle$.
Thus elements of the fundamental group are words generated by these distinct loops.
This fundamental group will map to a subgroup $H$ of $\pi_1( S^1 \vee S^1) \cong \mathbb{Z} * \mathbb{Z}$ under the covering map.
With this covering map, I think, $a \mapsto a$, $bab^{-1} \mapsto a$ and $b^2 \mapsto b^2$.
I can't quite put this information together to determine $H$. My guess would be $\langle a, b^2 \rangle$ but I am not quite sure if this is correct.
Any help is appreciated!
