The augmented Zagreb index (AZI) of a graph G is the sum of the weights d(u)d(v)/(d(u)+d(v)−2)3 over all edges uv of G, where d(u) is the degree of the vertex u in G. Let Ung be the set of all unicyclic graphs on n vertices with girth g. In this paper, we prove that if G is a graph with the maximum AZI in Ung, such that n≥32(g−2)+451 and g≥4, then G is a sunlet graph, i.e. G is a unicyclic graph such that removing all leaves gives a cycle. Furthermore, we have proven that G is always a good sunlet graph, in which all leaves are adjacent to the vertices in the cycle in a regular pattern. Finally we propose a new tool that could help to characterize the graph with the maximum AZI in Ung.