  If $m \in \mathbb{N}$ and $A$ is a finite subset of $\bigcup_{k \in
\mathbb{N} \setminus \{0,1\}} \{1,\ldots,m-1\}^k$, then we denote by
  \begin{align*}
  \mathscr{C}(m,A) = \left\{S\in \mathscr{S}_m \mid s_1+\cdots+s_k-m \in S
\mbox{ if } (s_1,\ldots,s_k)\in S^k \mbox{ and }\right. \\ \left.(s_1 \bmod m,
\ldots, s_k \bmod m)\in A \right\}.
  \end{align*}
  In this work we prove that $\mathscr{C}(m,A)$ is a Frobenius pseudo-variety.
We also show algorithms that allows us to establish whether a numerical
semigroup belongs to $\mathscr{C}(m,A)$ and to compute all the elements of
$\mathscr{C}(m,A)$ with a fixed genus. Moreover, we introduce and study three
families of numerical semigroups, called of second-level, thin and strong, and
corresponding to $\mathscr{C}(m,A)$ when $A=\{1,\ldots,m-1\}^3$,
$A=\{(1,1),\ldots,(m-1,m-1)\}$, and $A=\{1,\ldots,m-1\}^2 \setminus
\{(1,1),\ldots,(m-1,m-1)\}$, respectively.
