The degree/diameter problem for directed graphs is the problem of determining the largest possible order for a digraph with given maximum out-degree d and diameter k. An upper bound is given by the Moore bound M(d,k)=1+d+d^2+...+d^k$ and almost Moore digraphs are digraphs with maximum out-degree d, diameter k and order M(d,k)-1. In this paper we will look at the structure of subdigraphs of almost Moore digraphs, which are induced by the vertices fixed by some automorphism varphi. If the automorphism fixes at least three vertices, we prove that the induced subdigraph is either an almost Moore digraph or a diregular k-geodetic digraph of degree d'<d-1, order M(d',k)+1 and diameter k+1. As it is known that almost Moore digraphs have an automorphism r, these results can help us determine structural properties of almost Moore digraphs, such as how many vertices of each order there are with respect to r. We determine this for d=4 and d=5, where we prove that except in some special cases, all vertices will have the same order.