  Let $G$ be a graph that contains an induced subgraph $H$. A retraction from
$G$ to $H$ is a homomorphism from $G$ to $H$ that is the identity function on
$H$. Retractions are very well-studied: Given $H$, the complexity of deciding
whether there is a retraction from an input graph $G$ to $H$ is completely
classified, in the sense that it is known for which $H$ this problem is
tractable (assuming $\mathrm{P}\neq \mathrm{NP}$). Similarly, the complexity of
(exactly) counting retractions from $G$ to $H$ is classified (assuming
$\mathrm{FP}\neq \#\mathrm{P}$). However, almost nothing is known about
approximately counting retractions. Our first contribution is to give a
complete trichotomy for approximately counting retractions to graphs of girth
at least $5$. Our second contribution is to locate the retraction counting
problem for each $H$ in the complexity landscape of related approximate
counting problems. Interestingly, our results are in contrast to the situation
in the exact counting context. We show that the problem of approximately
counting retractions is separated both from the problem of approximately
counting homomorphisms and from the problem of approximately counting list
homomorphisms --- whereas for exact counting all three of these problems are
interreducible. We also show that the number of retractions is at least as hard
to approximate as both the number of surjective homomorphisms and the number of
compactions. In contrast, exactly counting compactions is the hardest of all of
these exact counting problems.
