The nature of heterophilous graphs is significantly different from that of homophilous graphs, which causes difficulties in early graph neural network (GNN) models and suggests aggregations beyond the one-hop neighborhood. In this article, we develop a new way to implement multiscale extraction via constructing Haar-type graph framelets with desired properties of permutation equivariance, efficiency, and sparsity, for deep learning tasks on graphs. We further design a graph framelet neural network model permutation equivariant graph framelet augmented network (PEGFAN) based on our constructed graph framelets. The experiments are conducted on a synthetic dataset and nine benchmark datasets to compare the performance with other state-of-the-art models. The result shows that our model can achieve the best performance on certain datasets of heterophilous graphs (including the majority of heterophilous datasets with relatively larger sizes and denser connections) and competitive performance on the remaining.