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 intersect each other and so 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 parallel automorphism lines of the groups of the same cardinality do exist? Can this number vary in different forcing extensions?Precisely, define the equivalence relation $\sim$ on (isomorphism type of) 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 $\kappa$?

**Remark.** If an answer to the above question is in hand, the fact that every automorphism line eventually terminates actually gives us the number of groups which can arise as the *terminating point* of the automorphism line of a group of size $\kappa$ because two lines are parallel if and only if they have distinct terminating points.