Multiple group modeling is one of the methods to address the measurement noninvariance issue. Traditional studies on multiple group modeling have mainly focused on item responses. In computer-based assessments, joint modeling of response times and action counts with item responses helps estimate the latent speed and action levels in addition to latent ability. These two new data sources can also be used to further address the measurement noninvariance issue. One challenge, however, is to correctly model action counts which can be underdispersed, overdispersed, or equidispersed in real data sets. To address this, we adopted the Conway-Maxwell-Poisson distribution that accounts for different types of dispersion in action counts and incorporated it in the multiple group joint modeling of item responses, response times, and action counts. Bayesian Markov Chain Monte Carlo method was used for model parameter estimation. To illustrate an application of the proposed model, an empirical data analysis was conducted using the Programme for International Student Assessment (PISA) 2015 collaborative problem-solving items where potential measurement noninvariance issue existed between gender groups. Results indicated that Conway-Maxwell-Poisson model yielded better model fit than alternative count data models such as negative binomial and Poisson models. In addition, response times and action counts provided further information on performance differences between groups.