The spread of information through socio-technical systems determines which individuals are the first to gain access to opportunities and insights. Yet, the pathways through which information flows can be skewed, leading to systematic differences in access across social groups. These inequalities remain poorly characterized in settings involving nonlinear social contagion and higher-order interactions that exhibit homophily. We introduce a enerative model for hypergraphs with hyperedge homophily, a hyperedge size-dependent property, and tunable degree distribution, called the $\texttt{H3}$ model, along with a model for nonlinear social contagion that incorporates asymmetric transmission between in-group and out-group nodes. Using stochastic simulations of a social contagion process on hypergraphs from the $\texttt{H3}$ model and diverse empirical datasets, we show that the interaction between social contagion dynamics and hyperedge homophily -- an effect unique to higher-order networks due to its dependence on hyperedge size -- can critically shape group-level differences in information access. By emphasizing how hyperedge homophily shapes interaction patterns, our findings underscore the need to rethink socio-technical system design through a higher-order perspective and suggest that dynamics-informed, targeted interventions at specific hyperedge sizes, embedded in a platform architecture, offer a powerful lever for reducing inequality.