  For any $\epsilon>0$ and $n>(1+\epsilon)t$, $n>n_0(\epsilon)$ we determine
the size of the largest $t$-intersecting family of permutations, as well as
give a sharp stability result. This resolves a conjecture of Ellis, Friedgut
and Pilpel (2011) and shows the validity of conjectures of Frankl and Deza
(1977) and Cameron (1988) for $n>(1+\epsilon )t$. We note that, for this range
of parameters, the extremal examples are not necessarily trivial, and that our
statement is analogous to the celebrated Ahlswede-Khachatrian theorem. The
proof is based on the refinement of the method of spread approximations,
recently introduced by Kupavskii and Zakharov (2022).
