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Let ${\mathcal M}_2(\mathbb F)$ be the algebra of 2$\times$2 matrices overthe real or complex field $\mathbb F$. For a given positive integer $k\geq 1$,the $k$-commutator of $A$ and $B$ is defined by $[A,B]_k=[[A,B]_{k-1},B]$ with$[A,B]_0=A$ and $[A,B]_1=[A,B]=AB-BA$. The main result is shown that a map$\Phi: {\mathcal M}_2(\mathbb F)\to {\mathcal M}_2(\mathbb F)$ with rangecontaining all rank one matrices satisfies that $[\Phi(A),\Phi(B)]_k = [A,B]_k$ for all $A, B\in{\mathcal M}_2(\mathbb F)$ if and only if there exist afunctional $h :{\mathcal M}_2(\mathbb F) \rightarrow {\mathbb F}$ and a scalar$\lambda \in{\mathbb F}$ with $\lambda^{k+1} = 1$ such that $\Phi(A) = \lambdaA + h(A)I$ for all $A \in{\mathcal M}_2(\mathbb F)$.

Strong $k$-commutativity preserving maps on 2$\times$2 matrices