We propose and characterize protocols to engineer exact quantum revivals, discrete time crystals (DTCs), and entanglement oscillations in the periodically driven central spin model. While period-doubling DTCs have been observed in this system before, we uncover a unifying interaction-induced echo mechanism underlying several distinct dynamical regimes. We first demonstrate that this echo can enable exact period-doubling revivals when the Ising interaction strength, $\lambda$, between the central spin and the $N_{\mathrm{sat}}$ satellite spins is tuned to $2\pi$. Notably, these revivals persist for arbitrary $N_{\mathrm{sat}}$ and transverse field $g$. This not only stabilizes the DTC response over a wide parameter regime, but also leads to a dynamical freezing regime. Furthermore, when $\lambda =(2m+1)\pi$ and $g=(2n+1)\pi /2$ ($\forall m,n\in \mathbb{Z}$), this echo induces a Clifford group structure. Consequently, higher-period revivals emerge that naturally steer the system through an entangled manifold of Bell-cat and spin-cat states. We establish that due to parity-dependent phases accumulated during the echo, the recurrence period is $12T\left(24T\right)$ for even (odd) $N_{\mathrm{sat}}$. Finally, we demonstrate that the multipartite entanglement generated by the Clifford dynamics can be harnessed for multiparameter metrology, with odd $N_{\mathrm{sat}}$ enabling Heisenberg-limited sensitivity.

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• Entanglement manipulation
• Quantum metrology
• Time crystals
• Floquet systems
• Quantum spin models