Group isomorphism and homomorphism are core concepts in abstract algebra, and student understanding of isomorphism has received extensive attention in line with the centrality of this topic. However, limited work has directly examined student conceptions of homomorphism or what metaphors students use to express their thought processes while problem solving. Based on interviews with four students, we contrast two students who used predominantly formal definition and mapping-centered metaphors for homomorphism with two who additionally used sameness-centered metaphors and note that the usage or non-usage of sameness-centered metaphors was not indicative of successful problem solving. Implications include the alignment between students’ metaphors and those used in instruction, indicating the importance of attending to metaphors when teaching, and the importance of discussing what is intended by some sameness-based metaphors, such as operation-preservation. •Students used sameness metaphors for isomorphism tasks and were successful.•All students used mapping and formal definition cluster metaphors for homomorphism.•Two students additionally used sameness-based metaphors for homomorphism.•Most student metaphors had arisen in class previously.•Not all students took up all metaphors that had been used in their class.