Clustered data are frequently subject to missing values, especially those collected from longitudinal studies. The main focus of the analysts is usually not on the clustering variables, hence the group-specific parameters are treated as nuisance. If a fixed effects formulation is preferred and the total number of clusters is large relative to the single-group sizes, classical frequentist techniques are often misleading. We propose here to combine multiple imputation and the modified profile likelihood function to obtain accurate inferences on a parameter of interest under models with incidental parameters for incomplete grouped observations. Such solution is examined via simulation studies which shed light on the convenience for the imputation model to take into account the clustered structure of the data.