This work deals with inferential aspects of INARCH(1) models which are adequate for overdispersed time series of counts. Besides, two approaches to approximate the marginal stationary distributions are analysed, namely, a Markov chain approach and the Poisson-Charlier expansion. As a possible application for these marginal probabilities, the computation of average run lengths (ARLs) of control charts is discussed. As the estimation of parameters is concerned, the maximum likelihood, conditional least squares and the moment estimators are derived, as well as some related asymptotic results. The first two estimators are used to construct several simultaneous confidence regions, whose finite sample performances are analysed through a simulation study. Finally, the model is used to analyse monthly strike data in the US during the period from January 1994 to December 2002. The model proved fit better to the data than four versions of the INAR(1) model.