= Solution
Let $F_2=\langle a,b\rangle=\pi_1(S^1\vee S^1)$ and take the connected two-sheeted <covering graph> corresponding to the kernel of
$$
F_2\longrightarrow C_2,\qquad a\longmapsto1,\quad b\longmapsto0.
$$
It has two vertices, a $b$-loop at each vertex, and two lifted $a$-edges joining the vertices. Choose one lifted $a$-edge as a spanning tree. The other three edges give the based loops
$$
\boxed{b,\qquad a^2,\qquad aba^{-1}}.
$$
Collapsing the spanning tree identifies the cover with a wedge of three circles, so these loops freely generate its fundamental group. Hence they generate an explicit subgroup
$$
\boxed{\langle b,a^2,aba^{-1}\rangle\le F_2}
$$
isomorphic to $F_3$, in agreement with the <Reidemeister–Schreier theorem>.
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