Coded aperture imaging is an emerging technique allowing thin form factor cameras that can be cheaply constructed. Many applications benefit from using such lensless cameras, such as face recognition. We propose a method for face recognition using coded aperture images that does not require retraining any component of the face recognition pipeline, but instead applies post-processing to the images with deep learning refinement so that they are compatible with existing face recognition for RGB images. We generate training data with a simulation process, based on the convolutional model of a lensless camera, and train a neural network to reconstruct face images. We train our network with a multi-term loss function to refine identity information in the reconstructed face image. We provide extensive experiments on different face recognition datasets, including LFW, CA-LFW, CP-LFW, AgeDB, FERET, and FRGC, showing the effectiveness and generalization of our proposed method. Our source code will be made available publicly to facilitate the reproducibility of our work.