= Solution
Take the <sphere> double cover $S^2\to\mathbb{RP}^2$ and denote the two points above the wedge point by $v_0,v_1$. Attach one $x$ <circle> at each point, mapping identically onto the <circle> summand. At both points there is one incoming and one outgoing $x$ branch, so this is a <connected> two-sheeted cover of the wedge.
The <sphere> is simply <connected>, while each attached <circle> supplies one generator. Join the two attachment points by an arc in the <sphere> and collapse that contractible arc; the <fundamental group> is the <free product> of the two <circle> groups, hence free of rank two. A <lift> of $y$ joins $v_0$ to $v_1$. The two loop generators therefore project to
$$
\boxed{x,\qquad yxy^{-1}=yxy}.
$$
Injectivity of the <covering map> identifies their subgroup with the \b[<free group> on these two generators].
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