  If $f : \tilde{\Gamma} \rightarrow \Gamma$ is a covering map between
connected graphs, and $H$ is the subgroup of $\pi_1(\Gamma,v)$ used to
construct the cover, then it is well known that the group of deck
transformations of the cover is isomorphic to $ N(H)/H$, where $N(H)$ is the
normalizer of $H$ in $\pi_1(\Gamma,v)$. We show that an entirely analogous
result holds for immersions between connected graphs, where the subgroup $H$ is
replaced by the closed inverse submonoid of the inverse monoid $L(\Gamma,v)$
used to construct the immersion. We observe a relationship between group
actions on graphs and deck transformations of graph immersions. We also show
that a graph immersion $f : \tilde{\Gamma} \rightarrow \Gamma$ may be extended
to a cover $g : \tilde{\Delta} \rightarrow \Gamma$ in such a way that all deck
transformations of $f$ are restrictions of deck transformations of $g$.
