Orbit decidability and the conjugacy problem for some extensions of groups

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Given a short exact sequence of groups with certain conditions, 1→ F→ G→ H→ 1, we prove that 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 Z² Fₘ, F₂ Fₘ, Fₙ Z, and Zⁿ ₐ Fₘ with virtually solvable action group A GLₙ(Z). Also, we give an easy way of constructing groups of the form Z⁴ Fₙ and F₃ Fₙ 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(F₂) is given.