AbstractExtremal problems for graph homomorphisms have recently become a topic of much research. Let denote the number of homomorphisms from G to H. A natural set of problems arises when we fix an image graph H and determine which graph(s) G on n vertices and m edges maximize . We prove that if H is loop‐threshold, then, for every n and m, there is a threshold graph G with n vertices and m edges that maximizes . Similarly, we show that loop‐quasi‐threshold image graphs have quasi‐threshold extremal graphs. In the case , the path on three vertices in which every vertex in looped, the authors [5] determined a set of five graphs, one of which must be extremal for . Also in this article, using similar techniques, we determine a set of extremal graphs for “the fox,” a graph formed by deleting the loop on one of the end‐vertices of . The fox is the unique connected loop‐threshold image graph on at most three vertices for which the extremal problem was not previously solved.