Orbit decidability and the conjugacy problem for some extensions of groups

Given a short exact sequence of groups with certain conditions, $1\to F\to G\to H\to 1$,weprovethat$G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A⩽Aut(F)$ is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form ${\mathbb{Z}}^2⋊F_m$, $F_2⋊F_m$, $F_n \rtimes $\mathbb{Z}$$,and$$\mathbb{Z}$^n \rtimes_A F_m$ with virtually solvable action group $A⩽G{L}_n(\mathbb{Z})$. Also, we give an easy way of constructing groups of the form ${\mathbb{Z}}^4⋊F_n$ and $F_3⋊F_n$ with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in $Aut(F_2)$ is given.