  In this paper we raise the realisability problem in arrow categories. Namely,
for a fixed category $\mathcal{C}$ and for arbitrary groups $H\le G_1\times
G_2$, is there an object $\phi \colon A_1 \rightarrow A_2$ in
$\operatorname{Arr}(\mathcal{C})$ such that
$\operatorname{Aut}_{\operatorname{Arr}(\mathcal{C})}(\phi) = H$,
$\operatorname{Aut}_{\mathcal{C}}(A_1) = G_1$ and
$\operatorname{Aut}_{\mathcal{C}}(A_2) = G_2$? We are interested in solving
this problem when $\mathcal C =\mathcal{H}oTop_*$, the homotopy category of
pointed topological spaces. To that purpose, we first settle that question in
the positive when $\mathcal C = \mathcal{G}raphs$.
  Then, we construct an almost fully faithful functor from $\mathcal{G}raphs$
to $\operatorname{CDGA}$, the category of commutative differential graded
algebras, that provides among other things, a positive answer to our question
when $\mathcal C = \operatorname{CDGA}$ and, as long as we work with finite
groups, when $\mathcal C =\mathcal{H}oTop_*$. Some results on representability
of concrete categories are also obtained.
