Consider a one-sided topological Markov chain \((\Sigma_{A},\theta)\) on a finite or countable alphabet and a continuous function \(\varphi:\Sigma_{A}\to (0,\infty)\). Denote the Gurevič pressure of \((\Sigma_{A},\theta,\varphi)\) by \(P_{G}(\theta,\varphi)\). For a countable discrete group \(G\) and a map \(\psi:\Sigma_{A}\to G\) define the group extension \(T:\Sigma_{A}\times G\to\Sigma_{A}\times G\) by \(T(x,g):=(\theta(x),g\cdot\psi(x))\). If \(\varphi\) is weakly symetric, \(P_{G}(\theta,\varphi)\) is finite, \(\theta\) is topologically mixing, \(T\) is topologically transitive, and \(G\) is amenable, then \(P_{G}(T,\varphi)=P_{G}(\theta,\varphi)\). Under certain conditions on \(\theta\) and \(\varphi\) it is proved that \(P_{G}(T,\varphi)=P_{G}(\theta,\varphi)\) implies that \(G\) is amenable. This is used to prove that a normal subgroup \(N\) of an essentially free Kleinian group \(G\) has the same exponent of convergence as \(G\) if and only if \(G/N\) is amenable.