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We generalise Savchenko's definition of topological entropy for special flows over countable Markov shifts by considering the corresponding notion of topological pressure. For a large class of Hölder continuous height functions not necessarily bounded away from zero, this pressure can be expressed by our new notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words, and we are able to set up a variational principle in this context. Investigating the dependence of induced pressure on the subset of words, we give interesting new results connecting the Gurevič and the classical pressure with exhaustion principles for a large class of Markov shifts. In this context we consider dynamical group extensions to demonstrate that our new approach provides a useful tool to characterise amenability of the underlying group structure.