Automorphisms of some cyclic extensions of free groups of rank three

Description of the group of outer automorphisms of the Gersten group was obtained by the author together with F. Dudkin in 2021 [7]. In this paper, we study the possibility of extending the methods of that work to an infinite class of cyclic extensions of a free group of rank three
$$G_k = \langle a, b, c, t | a^t = a, b^t = ba^k, c^t = c \rangle.$$
We have found the generating elements of the group $Out(G_k)$ and obtained a description of the structure of this group.