Consider a finitely presented group $G$ with presentation $P$ given by $\left\langle g_1,\ldots,g_n|\, r_1,\ldots,r_m\right\rangle$, equipped with a homomorphism $\rho\colon\, G\to H$ to a finitely generated group $H$ (generators in the presentation are ordered). Perform a sequence of Tietze transformations starting at $P$ and ending at presentation $P^\prime$ given by $\left\langle g^\prime_1,\ldots,g^\prime_n|\, r^\prime_1,\ldots,r^\prime_m\right\rangle$ which differs from $P$ by relabeling of generators (so $g_i\mapsto g_i^\prime$ for all $i$ is an isomorphism of presentations). Is it true that $\rho(g_i)=\rho(g_i^\prime)$ for all $i$ up to inner automorphism of $H$?

At first I thought this was trivial, but I'm worried that it might actually be false- if so, is there some way to measure such a failure of simple-connectedness of "presentations of $G$ over $H$".

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