Some classes of $H$-permutable and exactly $H$-permutable $n$-groupoids and $n$-quasigroups are considered. If $(Q,f)$ is an $n$-groupoid and $H$ a subgroup of the symmetric group $S_{n+1}$, such that for every $\sigma\in H$ $f(x_{\sigma_1},\dots,x_{\sigma_n}=x_{\sigma_{n+1}}\Leftrightarrow f(x_1,\dots,x_n)=x_{n+1}$ for all $x_1,\dots,{n+1}\in Q$, then $(Q,f)$ is called $H$-permutable. Moreover, if $H$ consists of all Bermutations with the given property, then $(Q,f)$ is exactly $H$-permutable. It is proved that every $H$-permutable $n$-groupoid is an $n$-quasigroup iff $H$ is a transitive permutation group. For some groups $H$ a class of exactly $H$-permutable $n$-quasigroups of any prime order $p>n+1$ is constructed, which establishes a conjecture of Hoffman [2] for such groups $H$. A corollary of this is the existence of $n$-quasigroups of every order $>n+1$ with exactly $\frac{(n+1)!}{n_1!\dots n_k!} $ conjugacy classes, where $n_1,\dots n_k$ are arbitrary positive integers such that $n_1+\dots+n_k\leq n+1$. The existence of exactly eyclic $n$-quasigroups of prime order $p=1$ $(\mod (n+1))$ and infinite order is also proved.