Summary: In this paper, we consider the sparse phase retrieval problem, recovering an \(s\)-sparse signal \(\mathfrak{x}^\natural\in\mathbb{R}^n\) from \(m\) phaseless samples \(y_i=|\langle\mathfrak{x}^\natural,\mathfrak{a}_i\rangle|\) for \(i=1,\dots,m\). Existing sparse phase retrieval algorithms are usually first-order and hence converge at most linearly. Inspired by the hard thresholding pursuit (HTP) algorithm in compressed sensing, we propose an efficient second-order algorithm for sparse phase retrieval. Our proposed algorithm is theoretically guaranteed to give an exact sparse signal recovery in finite (in particular, at most \(O(\log m+\log(\| \mathfrak{x}^\natural \|_2/| x_{\min}^\natural|))\) steps, when \(\{\mathfrak{a}_i\}_{i=1}^m\) are i.i.d. standard Gaussian random vector with \(m\sim O(s\log(n/s))\) and the initialization is in a neighborhood of the underlying sparse signal. Together with a spectral initialization, our algorithm is guaranteed to have an exact recovery from \(O(s^2\log n)\) samples. Since the computational cost per iteration of our proposed algorithm is the same order as popular first-order algorithms, our algorithm is extremely efficient. Experimental results show that our algorithm can be several times faster than existing sparse phase retrieval algorithms.