We will introduce the notion of $G$-Higgs bundle and focus in a particular case which gives us much more information, that of $G$ being the isometry group of a Hermitian symmetric space. In that case, Milnor Wood inequality will bound the Toledo invariant, and when $G$ is of tube type as well, the generalized Cayley correspondence will relate the moduli space for maximal Toledo invariant to a moduli space of $K2$-pairs over a reduced group. This result has been proved for the classical cases and we will deal the exceptional cases (related to groups of type E6 and E7).