We know that a chain map $f$ between chain complexes $(A_\bullet, d_{A,\bullet})$ and $(B_\bullet, d_{B,\bullet})$ induces homomorphisms between the homology groups:
$$(f_\bullet)_*:H_\bullet(A_\bullet, d_{A,\bullet})\to H_\bullet(B_\bullet, d_{B,\bullet}) $$
https://en.wikipedia.org/wiki/Chain_complex#Chain_maps
Suppose we have another chain map $g$ between chain complexes $(A_\bullet, \tilde{d}_{A,\bullet})$ and $(B_\bullet, \tilde{d}_{B,\bullet})$. That is, the modules $A_\bullet$ and $B_\bullet$ are the same, but the differentials are changed.
We have another (family of) induced homomorphisms: $$(g_\bullet)_*:H_\bullet(A_\bullet, \tilde{d}_{A,\bullet})\to H_\bullet(B_\bullet, \tilde{d}_{B,\bullet}) $$
Suppose further that we have $$H_n(A_\bullet,d_{A,\bullet})\cong H_n(A_\bullet,\tilde{d}_{A,\bullet})$$ and $$H_n(B_\bullet,d_{B,\bullet})\cong H_n(B_\bullet,\tilde{d}_{B,\bullet})$$ for all $n$.
Can we say anything about $\text{Im} (f_\bullet)_*$ and $\text{Im} (g_\bullet)_*$? In particular, is it true (or false) that $$\text{Im} (f_n)_*\cong \text{Im} (g_n)_*$$ for all $n$?
Thanks a lot for any help.
No, it's not true. There's no relationship between $f$ and $g$ in your hypotheses, so there is no reason to expect their images to be related.
For a counterexample, you can even keep the same differentials. Consider $A_0 = \mathbb{Z}$, $A_n = 0$ for $n \neq 0$, the differential $d_A$ is zero. Let $(B_\bullet, d_B)$ be the same chain complex. Let also $\tilde{d}_A = d_A$, $\tilde{d}_B = d_B$. Finally let $f : A \to B$ be the identity, and $g : A \to B$ be the zero map. Then $\operatorname{im}(f_*) = \mathbb{Z}$ in degree zero, whereas $\operatorname{im}(g_*) = 0$.