In isomorphism problems one often has a pair $(x,y)$ of objects in a category $\mathcal{C}$ and wants to describe the set $\mathrm{iso}(x,y)$ of isomorphisms $x\to y$ in the category. This is impractical as $|\mathrm{iso}(x,y)|$ can be exponentially larger than the data used to describe $x$ and $y$. The remedy:
$$\mathrm{iso}(x,y)=\mathrm{aut}(x)\phi$$
where $\mathrm{aut}(x)=\mathrm{iso}(x,x)$ is the group of automorphisms of $x$ and $\phi:x\to y$ is any element of $\mathrm{iso}(x,y)$. In general a group can be prescribed by a set of generators which is logarithmic in the order of the group (for countable infinite groups a finite generating sets would be your cut down).
Well in the expositions one wants to say what you are doing is a making $\mathrm{iso}(x,y)$ a coset of $\mathrm{aut}(x)$ -- it appeals to things already known and trusted. But that doesn't really make sense as its not inside some larger group. Rather it is inside a groupoid -- the groupoid of all invertible morphisms in $\mathcal{C}$. I see that groupoids give names to $\mathrm{aut}(x)$, things like automorphism group or Wikipedia says vertex groups.
Is it permissible to call $\mathrm{iso}(x,y)$ a coset in the groupoid? If not, is there a name for $\mathrm{iso}(x,y)$ other than "hom-set in the groidoid"?
If you're looking for group-theoretic points of view on the situation, I think what you're looking for is that, if it's nonempty, then $\mathrm{iso}(x,y)$ is an $\mathrm{aut}(x)$-torsor.