Conjugacy of pointwise conjugate homomorphisms

Given groups $H$ and $G$ , let us say that the pair $(H,G)$ has property (*) if any two injective homomorphisms $\alpha_0,\alpha_1\colon H\to G$ are conjugate if (and only if) they are pointwise conjugate.

Problem 1: Describe the class $C_1$ of groups $H$ such that $(H,G)$ has (*) for every group $G$ .
Problem 2: Describe the class $C_2$ of groups $G$ such that $(H,G)$ has (*) for every group $H$ .

Definitions: Given a group $G$ , two elements $a,b\in G$ are conjugate if there exists $x\in G$ such that $a=xbx^{-1}$ . Two homomorphisms $\alpha_0,\alpha_1\colon H\to G$ are pointwise conjugate if $\alpha_0(a)$ is conjugate to $\alpha_1(a)$ in $G$ for every $a\in H$ . (Thus, for each $a\in H$ there exists $x_a\in G$ such that $\alpha_0(a)=x_a\alpha_1(a)x_a^{-1}$ .) Further, $\alpha_0$ and $\alpha_1$ are conjugate if there exists $x\in G$ such that $\alpha_0(a)=x\alpha_1(a)x^{-1}$ for every $a\in H$ . If $\alpha_0$ and $\alpha_1$ are conjugate, then they are locally conjugate. Property (*) records that the converse holds (for injective homomorphisms).

Motivation: Problem 30 of the Scottish Book [1] (which is still open) asks to determine which groups $G$ have the following property: Pairs $(a,a')$ and $(b,b')$ in $G$ are conjugate (there exists $x\in G$ such that $a=xbx^{-1}$ and $a'=xb'x^{-1}$ ) if (and only if) for every word $w$ in two noncommuting variables the elements $w(a,a')$ and $w(b,b')$ are conjugate. Using property (*) formulated above, Problem 30 asks to determine which groups $G$ have the property that $(H,G)$ satisfies (*) for every subgroup $H$ of $G$ that is generated by two elements. The above Problem 2 is a more general (and possibly more natural) version of this problem.

An automorphism $\alpha\colon G\to G$ is class-preserving if $\alpha(a)$ is conjuagte to $a$ for every $a\in G$ . If $(G,G)$ has (*), then every class-preserving automorphism of $G$ is inner. The study of groups with (or without) outer class-preserving automorphisms has a long history; see for instance [2] and [3]. There exist finite groups with outer class-preserving automorphisms. In particular, there exist finite groups that belong neither to $C_1$ nor to $C_2$ . Note also that $C_2$ contains all abelian groups and that $C_1$ contains all cyclic groups.

[1]
R.D. Mauldin, The Scottish Book, Springer International Publishing, 2015. https://doi.org/10.1007/978-3-319-22897-6.
[2]
C.-H. Sah, Automorphisms of finite groups, Journal of Algebra. (1968) 47–68. https://doi.org/10.1016/0021-8693(68)90104-x.
[3]
M.K. Yadav, Class preserving automorphisms of finite p-groups: a survey, in: C.M. Campbell, M.R. Quick, E.F. Robertson, C.M. Roney-Dougal, G.C. Smith, G. Traustason (Eds.), Groups St Andrews 2009 in Bath, Cambridge University Press, 2007: pp. 569–579. https://doi.org/10.1017/cbo9780511842474.019.

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