Inner automorphism group of wreath product of Z2 and A5

Definition
This group is defined in the following ways:


 * 1) It is the defining ingredient::inner automorphism group of the defining ingredient::wreath product of Z2 and A5.
 * 2) The wreath product of Z2 and A5 has center isomorphic to defining ingredient::cyclic group:Z2, and this is a direct factor of the whole group. Our group is the other direct factor.
 * 3) It is the defining ingredient::external semidirect product of defining ingredient::elementary abelian group:E16 by alternating group:A5, where the latter acts on the former by taking its standard representation in characteristic two. See also modular representation theory of alternating group:A5.