Near 2-groups yield an isomorphism with many fixed points

For finite groups <f>G(∘)</f> and <f>G(&ast;)</f> define <f>d(∘,&ast;)</f> as the number of pairs <f>(u,v)∈G×G</f> with <f>u∘v≠u&ast;v</f>. If <f>|G|=n</f> is a power of two and <f>d(∘,&ast;)<n2/4</f>, then there exists an isomorphism <f>σ : G(∘)≅G(&ast;)</f> that fixes more than <f>7n/8</f> points of <f>G</f>. [Copyright &y& Elsevier]