On the number of autotopies of an $n$-ary qusigroup of order $4$

An algebraic system from a finite set $\mathrm{\Sigma }$ of cardinality $k$ and an $n$-ary operation $f$ invertible in each argument is called an $n$-ary quasigroup of order $k$. An autotopy of an $n$-ary quasigroup $(\mathrm{\Sigma },f)$ is a collection $({\theta }_0,{\theta }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\theta }_n)$ of $n+1$ permutations of $\mathrm{\Sigma }$ such that $f(\theta_1(x_1),…,\theta_n(x_n))\equiv \theta_0(f(x_1,\ldots,x_n))$\mathrm{.}Weshowthatevery$n$-aryquasigroupoforder$4$hasatleast$2^{[n/2]+2}$andnotmorethan$6\cdot 4^n$ autotopies. We characterize the $n$-ary quasigroups of order $4$ with $2^{(n+3)\mathrm{/}2}$, $2\cdot 4^n$,and$6\cdot 4^n$ autotopies.