I am trying to determine the number of $2:1$ coverings of of the wedge sum $S^1 \vee S^1 \vee S^1$. I know that proposition $1.32$ in Hatchers Algebraic Topology says that if $p:(X,x_0) \to (B,b_0)$ is a covering map, then the number of sheets of a covering is equal to the the index of the of $p_*(\pi_1(X,x_0))$ in $\pi_1(B,b_0)$. Thus, I think that this amounts to finding the number of index 2 subgroups in $\pi_1(S^1 \vee S^1 \vee S^1) \cong \mathbb{Z} *\mathbb{Z}*\mathbb{Z}$. Is this the correct approach or is there something else to consider.
Thank you