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 \documentclass[12pt]{minimal}
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\begin{document}$$1 \to G\prime \to G \to {{\mathbb{Z}}_{{p^k}}} \times \cdots \times {{\mathbb{Z}}_{{p^k}}},$$\end{document} where G′ ≅ ℤpk, and ζG/G′ is a direct factor of G/G′.Then G is a central product of an extraspecial pk-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 pk+l and pk+r, respectively, where 0 ≤ l, r ≤ k. Let AutG′G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G′, let AutG/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 AutG/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 → AutG′G → Aut G → Aut G′ → 1 is split. (ii) AutG′G/AutG/ζG,ζGG ≅ G1 × G2, where \documentclass[12pt]{minimal}
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\begin{document}$${\rm{Sp}}(2n - 2,{{\bf{Z}}_{{p^k}}})\ltimes H \le {G_1} \le {\rm{Sp}}(2n,{{\bf{Z}}_{{p^k}}})$$\end{document}, H is an extraspecial pk -group of order p(2n−1)k and \documentclass[12pt]{minimal}
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\begin{document}$$({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes \mathbb{Z}_{{p^k}}^{(m - 1)})\ltimes \mathbb{Z}_{{p^k}}^{(m)} \le {G_2} \le {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\ltimes \mathbb{Z}_{{p^k}}^{(m)}$$\end{document}. In particular, \documentclass[12pt]{minimal}
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\begin{document}$${G_1} = {\rm{Sp}}(2n - 2,{{\bf{Z}}_{{p^k}}})\ltimes H$$\end{document} if and only if l = k and r = 0; \documentclass[12pt]{minimal}
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\begin{document}$${G_1} = {\rm{Sp}}(2n,{{\bf{Z}}_{{p^k}}})$$\end{document} if and only if l ≤ r; \documentclass[12pt]{minimal}
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\begin{document}$${G_2} = ({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m - 1)})\ltimes {\mathbb{Z}}_{{p^k}}^{(m)}$$\end{document} if and only if r = k; \documentclass[12pt]{minimal}
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\begin{document}$${G_2} = {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m)}$$\end{document} if and only if r = 0. (iii) AutG′G/AutG/ζG,ζG/G′G ≅ G1 × G3, where G1 is defined in (ii); \documentclass[12pt]{minimal}
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\begin{document}$${\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m - 1)} \le {G_3} \le {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})$$\end{document}. In particular, \documentclass[12pt]{minimal}
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\begin{document}$${G_3}{\rm{= GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m - 1)}$$\end{document} if and only if r = k; \documentclass[12pt]{minimal}
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\begin{document}$${G_3} = {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})$$\end{document} if and only if r = 0. (iv) \documentclass[12pt]{minimal}
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\begin{document}$${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G/G\prime}}G \cong {\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G\rtimes {\mathbb{Z}}_{{p^k}}^{(m)}$$\end{document}. If m = 0, then \documentclass[12pt]{minimal}
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\begin{document}$${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G = {\rm{Inn}}\,G \cong \mathbb{Z}_{{p^k}}^{(2n)}$$\end{document}; If m > 0, then \documentclass[12pt]{minimal}
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\begin{document}$${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G \cong \mathbb{Z}_{{p^k}}^{(2nm)} \times \mathbb{Z}_{{p^{k - r}}}^{(2n)}$$\end{document}, and \documentclass[12pt]{minimal}
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\begin{document}$${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G/{\rm{Inn}}\,G \cong \mathbb{Z}_{{p^k}}^{(2n(m - 1))} \times \mathbb{Z}_{{p^{k - r}}}^{(2n)}$$\end{document}.
