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A decomposition \( \mathcal{D} \) of a graph \( H \) by a graph \( G \) is a partition of the edge set of \( H \) such that the subgraph induced by the edges in each part of the partition is isomorphic to \( G \). The intersection graph \( I(\mathcal{D}) \) of the decomposition \( \mathcal{D} \) has a vertex for each part of the partition and two parts \( A \) and \( B \) are adjacent if and only if they share a common node in \( H \). If \( I(\mathcal{D}) \cong H \), then \( \mathcal{D} \) is an automorphic decomposition of \( H \). If \( n(G) = \chi(H) \) as well, then we say that \( \mathcal{D} \) is a fully automorphic decomposition. In this paper, we examine the question of whether a fully automorphic host will have an even degree of regularity. We also give several examples of fully automorphic decompositions as well as necessary conditions for their existence.