  We introduce the 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. Firstly, the scaling function allows a direct access to
important thermodynamical quantities, which are usually given only implicitly
by certain identities involving the classically defined pressure. In this
context we generalise Savchenko's definition of entropy for special flows to a
corresponding notion of topological pressure and show that this new notion
coincides with the induced pressure for a large class of H\"older continuous
height functions not necessarily bounded away from zero. Secondly, the
dependence on the subset of words gives rise to interesting new results
connecting the Gurevi{\vc} and the classical pressure with exhausting
principles for a large class of Markov shifts. In this context we consider
dynamical group extentions to demonstrate that our new approach provides a
useful tool to characterise amenability of the underlying group structure.
