On finite groups all of whose bi-Cayley graphs of bounded valency are integral

Let k≥1 be an integer and Ik be the set of all finite groups G such that every bi-Cayley graph BCay(G,S) of G with respect to subset S of length 1≤|S|≤k is integral. Let k≥3. We prove that a finite group G belongs to Ik if and only if G≅Z3, Zr2 for some integer r, or S3.