Compounding Clifford's Theory

Let G be a group, N be a normal subgroup of G, and p be an irreducible (finite-dimensional) character of N in some algebraically closed field a. Assume that the stabilizer G, of p in G has finite index in G. Then Clifford's theory 2 gives us a central extension G of the multiplicative group F of a by GIN together with a one-to-one correspondence between the set Ch (G I A) of all irreducible characters of G having p as an N-constituent and the set Ch (G ) of all projective irreducible characters of GjIN corresponding to the extension G , i.e., of all irreducible characters of G having the natural embedding of F in ! as F-constituents. Now we add another normal subgroup K of G containing N and an irreducible character T of K having p as an N-constituent. We may take K to be the inverse image in G of Ky/N. Then K is a normal subgroup of G . Let T e Ch (K I q) correspond to * e Ch (K ) under the Clifford correspondence. One easily verifies that Ch (G I A) c Ch (G I9) corresponds to Ch(G I A). Furthermore, the stabilizers G and G, are related by (0.la) G o is the inverse image in G of (G , G,)/N. (0. lb) G, = (G, n G,)K. It follows that G : G + is finite if and only if G: G, is. When this happens, we may apply Clifford's theory to G, K, and A , obtaining an extension G of F by G,/K and a one-to-one correspondence between Ch (G I A) and Ch (G ). We may also apply Clifford's theory to G , K , and *, obtaining an extension G of F by G ,/K and a one-to-one correspondence between Ch (G I A) and Ch (G ). By (0.1) there is a natural isomorphism of G ,/K (Gfl GW)/K? onto G,/K. So G and G are extensions of F by isomorphic groups. Furthermore both