In the recently emerging field of group-based cryptography, the Conjugacy Search Problem (CSP) has gained traction as a non-commutative replacement of the Discrete Log Problem (DLP). The problem of finding a secure class of nonabelian groups for use as platforms is open and a subject of active research. This paper demonstrates a polynomial time solution of the CSP in an important class of nonabelian groups, the extraspecial $p$-groups. For this purpose, and as a further result, we provide a reduction of the CSP in certain types of central products. The consequences of our results are practically relevant for ruling out several groups as platforms, since several nonabelian groups are constructed by combining smaller groups by taking direct and central products.