Q: What is the "simplest" finite group $G$ for which we don't know how to realise it as a Galois group over $\mathbf{Q}$ ?

So here the word simplest might be interpreted in a broad sense. If you want something precise you may take the group of smallest order but I prefer to leave the question as it is. Also since naturally one classifies finite groups into families one may also ask the following

Q: What is the "simplest" example of a family of finite groups for which the inverse Galois problem is unknown?