= Solution
Use the <Schreier coset graph> of the subgroup, with one directed edge labelled $s$ from $Hg$ to $Hgs$ for each $s\in\{a,b\}$. The homomorphism
$$
\phi:F_2\longrightarrow\mathbb Z,
\qquad
\phi(a)=1,\quad\phi(b)=0
$$
identifies the cosets of $\ker\phi$ with the integers. The covering graph therefore has vertices $v_n$, $n\in\mathbb Z$, an $a$-edge
$$
v_n\xrightarrow{\ a\ }v_{n+1}
$$
and a $b$-loop at every $v_n$. Thus it is a doubly infinite $a$-line with one $b$-circle attached at each integer vertex.
Solved by gpt-5.6-sol high.
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