In future wireless networks, the synergy of non-orthogonal multiple access (NOMA) and cognitive radio (CR) can provide higher spectral efficiency and increased data rate. In this context, this article studies the performance of a space-time block code-aided overlay cognitive NOMA system based on Alamouti's (2 x 1, multiple-input single-output) code under the practical consideration of channel estimation errors (CEEs) and imperfect implementation of successive interference cancellation (SIC) at the receiver. We evaluate the closed-form expressions, after approximation, for the system throughput (ST) and outage probability (OP) over Rayleigh fading channels for both the secondary and primary networks. Asymptotic expressions of the OP and ST at high signal-to-noise ratio region are also evaluated to obtain further insights into the proposed scheme. Further, the article investigates the impact of various parameters on the OP and ST of the proposed scheme. Moreover, the OP of STBC orthogonal multiple access (OMA) based overlay cognitive network is also determined and compared with the presented network. Finally, simulations are used to validate the performance analysis of the proposed network and the accuracy of the derived analytical results. The simulation result shows that the presented system achieves significant gain in terms of the OP and ST compared to the STBC-OMA based overlay cognitive system and a recently proposed scheme under imperfect SIC and CEE conditions.