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In [A. Neri, P. Santonastaso, F. Zullo. Extending two families of maximumrank distance codes], the authors extended the family of $2$-dimensional$\mathbb{F}_{q^{2t}}$-linear MRD codes recently found in [G. Longobardi, G.Marino, R. Trombetti, Y. Zhou. A large family of maximum scattered linear setsof $\mathrm{PG}(1,q^n)$ and their associated MRD codes]. Also, for $t \geq 5$they determined equivalence classes of the elements in this new family andprovided the exact number of inequivalent codes in it. In this article, wecomplete the study of the equivalence issue removing the restriction $t \geq5$. Moreover, we prove that in the case when $t=4$, the linear sets of theprojective line $\mathrm{PG}(1,q^{8})$ ensuing from codes in the relevantfamily, are not equivalent to any one known so far.

On the equivalence issue of a class of $2$-dimensional linear Maximum Rank Distance codes