Abstract The twin group T n {T_{n}} is a right-angled Coxeter group generated by n - 1 {n-1} involutions, and the pure twin group PT n {\mathrm{PT}_{n}} is the kernel of the natural surjection from T n {T_{n}} onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in T n {T_{n}} , which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z-classes of involutions in T n {T_{n}} . We give a new proof of the structure of Aut ( T n ) {\operatorname{Aut}(T_{n})} for n ≥ 3 {n\geq 3} , and show that T n {T_{n}} is isomorphic to a subgroup of Aut ( PT n ) {\operatorname{Aut}(\mathrm{PT}_{n})} for n ≥ 4 {n\geq 4} . Finally, we construct a representation of T n {T_{n}} to Aut ( F n ) {\operatorname{Aut}(F_{n})} for n ≥ 2 {n\geq 2} .