= Solution
For $k>0$, the surface presentation
$$
\pi_1(\Sigma_2)=\langle a_1,b_1,a_2,b_2\mid[a_1,b_1][a_2,b_2]=1\rangle
$$
has an epimorphism to $\mathbb Z/k$ sending $a_1$ to $1$ and the other generators to zero. Its kernel has index $k$. By the <classification of connected covering spaces>, it determines a connected $k$-sheeted cover $p:\widetilde\Sigma\to\Sigma_2$. Pull back the orientation, so each local sheet has degree $+1$ and $\deg p=k$. Counting lifts of cells gives $\chi(\widetilde\Sigma)=k\chi(\Sigma_2)=-2k$, hence $2-2g=-2k$ and $g=k+1$.
For $k<0$, use the $|k|$-sheeted cover and compose with an <orientation-reversing diffeomorphism> of $\Sigma_2$, for instance reflection in a symmetric embedded genus-two surface. By <multiplicativity of mapping degree>, the degree becomes $-|k|=k$. For $k=0$ take a constant map from the torus. Thus the <arbitrary-degree maps to a surface of genus two> can be chosen with
$$
\boxed{g(k)=|k|+1,\qquad\Sigma_{|k|+1}\longrightarrow\Sigma_2\text{ of degree }k\quad(k\in\mathbb Z).}
$$
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