Let $F$ be an arbitrary field of characteristic zero, and let $M^{(m,k)}(F)$ be a matrix superalgebra over $F$. It is known from the theory of algebras with polynomial identities that the superalgebra $M^{(m,k)}(F)$ has a finite basis of $Z_2$-graded identities. Therefore, the problem of describing such a basis arises naturally. At the present moment of time, there is no such description. First of all, this is due to the fact that there are no effective methods for finding the usual or $Z_2$-graded identities of a superalgebra $M^{(m,k)}(F)$. Nevertheless, for some values of $m$, $k$, such identities can still be found. For this purpose, one uses either computer computations or the well-developed apparatus of the representation theory of the symmetric group $S_n$ and the general linear group $GL_p$. More precisely, to find $Z_2$-graded identities of a superalgebra $M^{(m,k)}(F)$ for small values of $m,k$, one studies the sequence $\{\chi_n\}$ of characters of representations of either groups $S_r\times S_{n-r}$ or group $GL_p\times GL_p$. For each such group, one constructs a vector $F$-space in the free algebra $F\{Y\bigcup Z\}$. At the same time, with respect to the action of group $S_r\times S_{n-r}$ ($GL_p\times GL_p$) on its vector space, it has the structure of a left $S_r\times S_{n-r}$ ($GL_p\times GL_p$) module. However, it turns out that it is computationally preferable to work with the characters representation sequence of the group $GL_p\times GL_p$. In this paper, we study the sequence of $GL_p\times GL_p$-characters $\{\chi_n\}$  of matrix superalgebra $M^{(2,2)}(F)$. This uses the fact that between pairs of partitions $(\lambda,\mu)$, where $\lambda\vdash r,\, \mu\vdash n-r$ and irreducible $GL_p\times GL_p$-modules, there is a one-to-one correspondence. Moreover, we investigate only those multiplicities in the decomposition of the character $\chi_n$ that are associated with irreducible $GL_p\times GL_p$-modules corresponding to pairs of partitions $(\lambda,\mu)$ of the form $(0,\mu)$. It is shown that if the height $h(\mu)$ of the Young diagram $D_\mu$ for a pair $(0,\mu)$ is no more than five, then the multiplicity $m_{0,\mu}$ of the irreducible $GL_p\times GL_p$-character $\chi_n$ is different from zero.