  In this paper we make an attempt to study right loops $(S, o)$ in which, for
each $y\in S$, the map $\sigma_y$ from the inner mapping group $G_S$ of $(S,
o)$ to itself given by $\sigma_y (h)(x) o\ h(y)= h(xoy)$, $x\in S, h\in G_S$ is
a homomorphism. The concept of twisted automorphisms of a right loop and also
the concept of twisted right gyrogroup appears naturally and it turns out that
the study is almost equivalent to the study of twisted automorphisms and a
twisted right gyrogroup. A representation theorem for twisted right gyrogroup
is established. We also study relationship between twisted gyrotransversals and
twisted subgroups (a concept which arose as a tool to study computational
complexity involving class NP).
