Let $(A,f)$ be an $n$-ary semigroup ($n \ge 2$). Then there exists a ($2$-ary) semigroup $(\overline A,*)$ with an inclusion homomorphism $A \hookrightarrow \overline A$ such that that the restriction of its $n$-ary derivation to $A$ coincides with $f$, i.e. $(*_n)|_A = f$.

Is this true?

One possible direction would be, $(A,f) \cong \mathcal F_n(A) / \sigma$, and then $(\overline A, *) = \mathcal F_2(A) / \langle\sigma\rangle$ with the usual inclusion $\mathcal F_n(A) \hookrightarrow \mathcal F_2(A)$ would satisfy the theorem, if one could show that $\langle\sigma\rangle \cap \mathcal F_n(A)^2 = \sigma$.

(Where, $\mathcal F_n(A)$ is the free $n$-ary semigroup of strings $s$ which have length $|s| \equiv 1 \bmod (n-1)$ with entries in $A$ and operation $n$-ary concatenation. And $\langle\sigma\rangle$ is the smallest congruence on $\mathcal F_2(A)$ containing $\sigma$.)

edit: The terms "$n$-ary semigroup" and "derived" are used in the sense of this paper: http://www.quasigroups.eu/contents/download/2006/14_14.pdf .