The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_n$, can be embedded in any central extension of $A_n$ if and only if $n\equiv 0\left(mod8\right)$, or $n\equiv 2\left(mod8\right)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$, every central extension of $A_n$ occurs as a Galois group over $\mathbf{Q}$.