Let p be an odd prime, and let k be a nonzero nature number. Suppose that nonabelian group G is a central extension as follows

where G′ ≅ ℤ_pk, and ζG/G′ is a direct factor of G/G′.Then G is a central product of an extraspecial p^k-group E and ζG. Let ∣E∣ = p^(2n+1)k and ∣ζG∣ = p^(m+1)k. Suppose that the exponents of E and ζG are p^k+l and p^k+r, respectively, where 0 ≤ l, r ≤ k. Let Aut_G′G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G′, let Aut_G/ζG,ζGG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζG, and let Aut_{G/ζG,ζG/G′}G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on ζG/G′. Then (i) The group extension 1 → Aut_G′G → Aut G → Aut G′ → 1 is split. (ii) Aut_G′G/Aut_G/ζG,ζGG ≅ G₁ × G₂, where \({\rm{Sp}}(2n - 2,{{\bf{Z}}_{{p^k}}})\ltimes H \le {G_1} \le {\rm{Sp}}(2n,{{\bf{Z}}_{{p^k}}})\), H is an extraspecial p^k -group of order p^(2n−1)k and \(({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes \mathbb{Z}_{{p^k}}^{(m - 1)})\ltimes \mathbb{Z}_{{p^k}}^{(m)} \le {G_2} \le {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\ltimes \mathbb{Z}_{{p^k}}^{(m)}\). In particular, \({G_1} = {\rm{Sp}}(2n - 2,{{\bf{Z}}_{{p^k}}})\ltimes H\) if and only if l = k and r = 0; \({G_1} = {\rm{Sp}}(2n,{{\bf{Z}}_{{p^k}}})\) if and only if l ≤ r; \({G_2} = ({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m - 1)})\ltimes {\mathbb{Z}}_{{p^k}}^{(m)}\) if and only if r = k; \({G_2} = {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m)}\) if and only if r = 0. (iii) Aut_G′G/Aut_{G/ζG,ζG/G′}G ≅ G₁ × G₃, where G₁ is defined in (ii); \({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m - 1)} \le {G_3} \le {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\). In particular, \({G_3}{\rm{= GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m - 1)}\) if and only if r = k; \({G_3} = {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\) if and only if r = 0. (iv) \({\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G/G\prime}}G \cong {\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G\rtimes {\mathbb{Z}}_{{p^k}}^{(m)}\). If m = 0, then \({\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G = {\rm{Inn}}\,G \cong \mathbb{Z}_{{p^k}}^{(2n)}\); If m > 0, then \({\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G \cong \mathbb{Z}_{{p^k}}^{(2nm)} \times \mathbb{Z}_{{p^{k - r}}}^{(2n)}\), and \({\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G/{\rm{Inn}}\,G \cong \mathbb{Z}_{{p^k}}^{(2n(m - 1))} \times \mathbb{Z}_{{p^{k - r}}}^{(2n)}\).

设p为奇素数，k为非零自然数。假设非贝尔群G是中心扩展如下

其中G ′ ≅ ℤ _p k，且ζG/G ′ 是G / G ′的直接因数。那么G是一个特异p ^k群E和ζG的中心积。设 ∣ E ∣ = p ^{(2 n +1) k}和 ∣ ζG ∣ = p ^{( m +1) k}。假设E和ζG的指数分别为p ^k+l和p ^k+r，其中 0 ≤l, r ≤ k。让AUT _{ģ '} G ^是AUT的正常子组G ^由AUT的所有元素的ģ其中所导出的子组平凡作用ģ '，让AUT _{G /ζG，ζG} ģ是AUT的正常子组G ^包括所有中央构的ģ这也平凡作用于中心ζG，并让AUT _{G /ζG，ζG/ G '} G ^是AUT的正常子组G ^包括所有中央构的ģ也平凡作用于ζG/ G'. 然后 (i) 群扩展 1 → Aut _{G ′} G → Aut G → Aut G ′ → 1 被分裂。(ii) Aut _{G ′} G /Aut _G/ζG,ζG G ≅ G ₁ × G ₂，其中\({\rm{Sp}}(2n - 2,{{\bf{Z}}_{{p^ k}}})\ltimes H \le {G_1} \le {\rm{Sp}}(2n,{{\bf{Z}}}_{{p^k}}})\)，H是一个额外的p ^ķ顺序的-group p ^{（2 ñ -1）ķ}和\(({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes \mathbb{Z}_{{p^k}}^{ (m - 1)})\ltimes \mathbb{Z}_{{p^k}}^{(m)} \le {G_2} \le {\rm{GL}}(m,{{\bf{ Z}}_{{p^k}}})\ltimes \mathbb{Z}_{{p^k}}^{(m)}\)。特别地，\({G_1} = {\rm{Sp}}(2n - 2,{{\bf{Z}}}_{{p^k}}})\ltimes H\)当且仅当l = k和r = 0; \({G_1} = {\rm{Sp}}(2n,{{\bf{Z}}_{{p^k}}})\)当且仅当l ≤ r；\({G_2} = ({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p ^k}}^{(m - 1)})\ltimes {\mathbb{Z}}_{{p^k}}^{(m)}\)当且仅当r = k；\({G_2} = {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k} }^{(m)}\)当且仅当r = 0。 (iii) Aut _{G ′} G/ Aut _{G/ζG,ζG/G ′} G ≅ G ₁ × G ₃，其中G ₁在 (ii) 中定义); \({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^ {(m - 1)} \le {G_3} \le {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\)。特别地，\({G_3}{\rm{= GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{ {p^k}}^{(m - 1)}\)当且仅当r = k ; \({G_3} = {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\)当且仅当r = 0。 (iv) \({\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G/G\prime}}G \cong {\ rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G\rtimes {\mathbb{Z}}_{{p^k}}^{(m)} \)。如果m = 0，则\({\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G = {\rm{Inn}}\,G \cong \mathbb{Z}_{{p^k}}^{(2n)}\) ; 如果m > 0，则\({\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G \cong \mathbb{Z}_{{p^k }}^{(2nm)} \times \mathbb{Z}_{{p^{k - r}}}^{(2n)}\)和\({\rm{Au}}{{\rm {t}}_{G/\zeta G,\zeta G}}G/{\rm{Inn}}\,G \cong \mathbb{Z}_{{p^k}}^{(2n(m - 1))} \times \mathbb{Z}_{{p^{k - r}}}^{(2n)}\)。