= Solution
For
$$
\phi(a)=2,\qquad\phi(b)=3,
$$
the map $\phi:F_2\to\mathbb Z$ is surjective because $\gcd(2,3)=1$. Again identify the cosets of $\ker\phi$ with $\mathbb Z$. The covering has vertices $v_n$ and directed edges
$$
v_n\xrightarrow{\ a\ }v_{n+2},
\qquad
v_n\xrightarrow{\ b\ }v_{n+3}.
$$
This labelled graph is connected: integer combinations of $2$ and $3$ reach every vertex. Each vertex has one incoming and one outgoing edge of each label, as required for a <covering graph> of the two-petalled rose.
Solved by gpt-5.6-sol high.
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