  Let $\mathbb{F}_q$ denote the finite field with $q=p^\lambda$ elements.
Maximum Rank-metric codes (MRD for short) are subsets of $M_{m\times
n}(\mathbb{F}_q)$ whose number of elements attains the Singleton-like bound.
The first MRD codes known was found by Delsarte (1978) and Gabidulin (1985).
Sheekey (2016) presented a new class of MRD codes over $\mathbb{F}q$ called
twisted Gabidulin codes and also proposed a generalization of the twisted
Gabidulin codes to the codes $\mathcal{H}_{k,s}(L_1,L_2)$. The equivalence and
duality of twisted Gabidulin codes was discussed by Lunardoni, Trombetti, and
Zhou (2018). A new class of MRD codes in $M_{2n\times 2n}(\mathbb{F}_q)$ was
found by Trombetti-Zhou (2018). In this work, we characterize the equivalence
of the class of codes proposed by Sheekey, generalizing the results known for
twisted Gabidulin codes and Trombetti-Zhou codes. In the second part of the
paper, we restrict ourselves to the case $L_1(x)=x$, where we present its right
nucleus, middle nucleus, Delsarte dual and adjoint codes. In the last section,
we present the automorphism group of $\mathcal{H}_{k,s}(x,L(x))$ and compute
its cardinality. In particular, we obtain the number of elements in the
automorphism group of the twisted Gabidulin codes.
