Explicit Isomorphism of Fundamental Groups for Homotopic Attaching Maps

Question

In Hatcher we find a proof that if $X$ and $Y$ are spaces, and $A\subseteq X$ is such that, given any continuous $f_0:X\to Z$ and homotopy $f_{A,t}:A\times[0,1]\to Z$ such that $f_0|_A=f_{A,t}|_{A\times\{0\}}$, there is a function homotopy $f_t:X\times[0,1]\to Z$ extending the two, then for any homotopic maps $f,g:A\to Y$, the spaces $X\sqcup_fY$ and $X\sqcup_gY$ are homotopy equivalent, where $X\sqcup_hY$ with $h:A\to Y$ is the quotient of $X\sqcup Y$ by $x\sim h(x)$. Therefore, $X\sqcup_fY$ and $X\sqcup_gY$ have the same fundamental group when they are path connected. However, no explicit isomorphism is given. Let's choose basepoints $b_f$ and $b_g$ as $[y]$ for $y\in Y$, or $[x]$ for $x\in X\setminus A$ under both equivalence relations, which hopefully simplifies things. Is it usually possible to give an explicit isomorphism? If so, how might this be done?