Constructing model-agnostic group equivariant networks, such as equitune (Basu et al., 2023b) and its generalizations (Kim et al., 2023), can be computationally expensive for large product groups. We address this problem by providing efficient model-agnostic equivariant designs for two related problems: one where the network has multiple inputs each with potentially different groups acting on them, and another where there is a single input but the group acting on it is a large product group. For the first design, we initially consider a linear model and characterize the entire equivariant space that satisfies this constraint. This characterization gives rise to a novel fusion layer between different channels that satisfies an invariance-symmetry (IS) constraint, which we call an IS layer. We then extend this design beyond linear models, similar to equitune, consisting of equivariant and IS layers. We also show that the IS layer is a universal approximator of invariant-symmetric functions. Inspired by the first design, we use the notion of the IS property to design a second efficient model-agnostic equivariant design for large product groups acting on a single input. For the first design, we provide experiments on multi-image classification where each view is transformed independently with transformations such as rotations. We find equivariant models are robust to such transformations and perform competitively otherwise. For the second design, we consider three applications: language compositionality on the SCAN dataset to product groups; fairness in natural language generation from GPT-2 to address intersectionality; and robust zero-shot image classification with CLIP. Overall, our methods are simple and general, competitive with equitune and its variants, while also being computationally more efficient.