A subgroup X of a finite group G is called ∗ ^ \ast -standard if X ~ = X / O ( X ) \tilde X = X/O(X) is quasisimple, Y = C G ( X ) Y = {C_G}(X) is tightly embedded in G and N G ( X ) = N G ( Y ) {N_G}(X) = {N_G}(Y) . This generalizes the notion of standard subgroups. Theorem. Let G be a finite group with O ( G ) = 1 O(G) = 1 . Suppose X is ∗ ^ \ast -standard in G and X ~ / Z ( X ~ ) ≅ L 2 ( 2 n ) , U 3 ( 2 n ) \tilde X/Z(\tilde X) \cong {L_2}({2^n}),{U_3}({2^n}) or Sz ( 2 n ) {\text {Sz}}({2^n}) . Assume X ⋪ G X \ntriangleleft G . Then O ( X ) = 1 O(X) = 1 and one of the following holds: ( i ) E ( G ) ≅ X × X ({\text {i}})\;E(G) \cong X \times X . ( ii ) X ≅ L 2 ( 2 n ) ({\text {ii}})\;X \cong {L_2}({2^n}) and E ( G ) ≅ L 2 ( 2 2 n ) , U 3 ( 2 n ) o r L 3 ( 2 n ) E(G) \cong {L_2}({2^{2n}}),{U_3}({2^n})\;or\;{L_3}({2^n}) . ( iii ) X ≅ U 3 ( 2 n ) ({\text {iii}})\;X \cong {U_3}({2^n}) and E ( G ) ≅ L 3 ( 2 2 n ) E(G) \cong {L_3}({2^{2n}}) . ( iv ) X ≅ Sz ( 2 n ) ({\text {iv}})\;X \cong {\text {Sz}}({2^n}) and E ( G ) ≅ Sp ( 4 , 2 n ) E(G) \cong {\text {Sp}}(4,{2^n}) . ( v ) X ≅ L 2 ( 4 ) ({\text {v}})\;X \cong {L_2}(4) and E ( G ) ≅ M 12 , A 9 , J 1 , J 2 , A 7 , L 2 ( 25 ) , L 3 ( 5 ) o r U 3 ( 5 ) E(G) \cong {M_{12}},{A_9},{J_1},{J_2},{A_7},{L_2}(25),{L_3}(5)\;or\;{U_3}(5) . ( vi ) X ≅ Sz ( 8 ) ({\text {vi}})\;X \cong {\text {Sz}}(8) and E ( G ) ≅ Ru E(G) \cong {\text {Ru}} (the Rudvalis group). ( vii ) X ≅ L 2 ( 8 ) ({\text {vii}})\;X \cong {L_2}(8) and E ( G ) ≅ G 2 ( 3 ) E(G) \cong {G_2}(3) . ( viii ) X ≅ SL ( 2 , 5 ) ({\text {viii}})\;X \cong {\text {SL}}(2,5) and G has sectional 2-rank at most 4. In particular, if G is simple, G ≅ M 12 , A 9 , J 1 , J 2 , Ru , U 3 ( 5 ) , L 3 ( 5 ) , G 2 ( 5 ) , o r 3 D 4 ( 5 ) G \cong {M_{12}},{A_9},{J_1},{J_2},{\text {Ru}},{U_3}(5),{L_3}(5),{G_2}(5), or\;{^3}{D_4}(5) .