We consider the problem of sparse phase retrieval from Fourier transform magnitudes to recover the <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-sparse signal vector and its support <inline-formula><tex-math notation="LaTeX">$\mathcal {T}$</tex-math></inline-formula>. We exploit extended support estimate <inline-formula><tex-math notation="LaTeX">$\mathcal {E}$</tex-math></inline-formula> with size larger than <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> satisfying <inline-formula><tex-math notation="LaTeX">$\mathcal {E} \supseteq \mathcal {T}$</tex-math></inline-formula> and obtained by a trained deep neural network (DNN). To make the DNN learnable, it provides <inline-formula><tex-math notation="LaTeX">$\mathcal {E}$</tex-math></inline-formula> as the union of equivalent solutions of <inline-formula><tex-math notation="LaTeX">$\mathcal {T}$</tex-math></inline-formula> by utilizing modulo Fourier invariances. Set <inline-formula><tex-math notation="LaTeX">$\mathcal {E}$</tex-math></inline-formula> can be estimated with short running time via the DNN, and support <inline-formula><tex-math notation="LaTeX">$\mathcal {T}$</tex-math></inline-formula> can be determined from the DNN output rather than from the full index set by applying hard thresholding to <inline-formula><tex-math notation="LaTeX">$\mathcal {E}$</tex-math></inline-formula>. Thus, the DNN-based extended support estimation improves the reconstruction performance of the signal with a low complexity burden dependent on <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>. Numerical results verify that the proposed scheme has a superior performance with lower complexity compared to local search-based greedy sparse phase retrieval and a state-of-the-art variant of the Fienup method.