It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. Hagglund constructed Blowup $$(K_4, C)$$ and Blowup(Prism, $$C_4)$$ . Based on these two graphs, Chen constructed infinite families of bridgeless cubic graphs $$M_{0,1,2, \ldots ,k-2, k-1}$$ which are obtained from cyclically 4-edge-connected and admitting Fulkerson-cover cubic graphs $$G_0,G_1,\ldots , G_{k-1}$$ by recursive process. He obtained that every graph in $$M_{0,1,2,3}$$ has a Fulkerson-cover and gave the open problem that whether every graph in $$M_{0,1,2, \ldots ,k-2, k-1}$$ has a Fulkerson-cover. In this paper, we solve this problem and prove that every graph in $$M_{0,1,2, \ldots ,k-2, k-1}$$ has a Fulkerson-cover.