A loop $(Q, \circ)$ is called Basarab loop if it is both a left and a right Basarab loop; $(x\circ yx^\rho)\circ xz= x\circ yz$ and $yx\circ (x^\lambda z\circ x)= yz \circ x$ hold for all $x, y, z\in Q$ respectively. In this paper, the characterizations of the Bryant-Schneider group of a Basarab loop are studied using the left and right Basarab loop identities. It is shown that the element, $x^\lambda~ (x^\rho)$ is in the left (right) nucleus if and only if the middle inner map $T_x$ (inverse $T_x^{-1}$) is an automorphism. It is revealed that every crypto-automorphism of a Basarab loop is an element of the Bryant-Schneider group. Some related algebraic properties were also characterized. Furthermore, elements of the Bryant-Schneider group of a Basarab loop in terms of pseudo-automorphism and automorphism are also characterized. A subgroup of the Bryant-Schneider group, characterized by the Basarab loop, is established. Finally, a right pseudo-automorphic characterization of the isotopy-isomorphy of a Basarab loop is carried out.