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](http://en.wikipedia.org/wiki/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 $H$".



Monodromy in presentations of one group over another
The question is effectively answered by HW and by Misha in the comments. Namely, a sequence of Tietze transformations starting and ending at "the same" presentation defines an automorphism of $G$. But the action of $\mathrm{Out}(G)$ on $\mathrm{Hom}(G,H)/\mathrm{Aut}(H)$ is non-trivial in general. 


