Given a group $G$, one can define the transfinite line of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the direct limit is taken at the limit stages:

$G\rightarrow Aut(G)\rightarrow Aut(Aut(G))\rightarrow\cdots\rightarrow G_{\alpha}\rightarrow G_{\alpha+1}\rightarrow\cdots$

The line terminates when a fixed point is reached, namely one of the groups in the chain is isomorphic to its automorphism group by the natural map. According to a result of Hamkins it is known that every automorphism line terminates. So there is no automorphism line of length $\text{Ord}$.

Definition. It is clear that the automorphism line of many non-isomorphic groups may have the same terminating point. In this case, we say that two automorphism lines have converged. Otherwise, we call them parallel. Precisely, the automorphism lines of $G$ and $H$ are convergent if there are ordinals $\alpha, \beta$ such that $G_{\alpha}\cong H_{\beta}$.

Question. How many distinct ending points can isomorphism lines of the groups of the same cardinality have (up to isomorphism)?

Precisely, define the equivalence relation $\sim$ on the groups so that $G\sim H$ if the automorphism lines of $G$ and $H$ converge. Let $\kappa$ be a (finite/infinite) cardinal and $\mathcal{C}_{\kappa}$ be the collection of all groups of size $\kappa$. What is the size of $\mathcal{C}_{\kappa}/\sim$ for different $\kappa$?