Steganography is a technique of hiding secret data in some unsuspected cover media so that it is visually imperceptible. The secret data as well as the cover media may be text or multimedia. Image steganography, where the cover media is an image, is one of the most commonly used schemes. Here, we focus on image steganography where the hidden data is also an image. Specifically, we embed grayscale secret images into a grayscale cover image, which is considered to be a challenging problem. Our goal is to develop a steganography scheme with enhanced embedding capacity while preserving the visual quality of the stego-image and ensuring that the stego-image is resistant to steganographic attacks. Our proposed scheme involves use of sparse approximation and our novel embedding rule, which helps to increase the embedding capacity and adds a layer of security. The stego-image is constructed by using the Alternating Direction Method of Multipliers (ADMM) to solve the Least Absolute Shrinkage and Selection Operator (LASSO) formulation of the underlying minimization problem. This method has a fast convergence, is easy to implement, and also is extensively used in image processing. Finally, the secret images are extracted from the constructed stego-image using the reverse of our embedding rule. Using these components together helps us to embed up to four secret images into one cover image (instead of the common embedding of two secret images) and forms our most novel contribution. We term our scheme SABMIS (Sparse Approximation Blind Multi-Image Steganography). We perform extensive experiments on several standard images, and evaluate the embedding capacity, Peak Signal-to-Noise Ratio (PSNR) value, mean Structural Similarity (MSSIM) index, Normalized Cross-Correlation (NCC) coefficients, entropy, and Normalized Absolute Error (NAE). We obtain embedding capacities of 2 bpp (bits per pixel), 4 bpp, 6 bpp, and 8 bpp while embedding one, two, three, and four secret images, respectively. These embedding capacities are higher than all the embedding capacities obtained in the literature until now. Further, there is very little deterioration in the quality of the stego-image as compared to its corresponding cover image (measured by above metrics). The quality of the original secret images and their corresponding extracted secret images is also almost the same. Further, due to our algorithmic design, our scheme is resistant to steganographic attacks as well.