  A systematic study of the total $\beta$-decay half-lives and $\beta$-delayed
neutron emission probabilities is performed. The $\beta$-strength function is
treated within the self-consistent density-functional + continuum-QRPA
framework including the Gamow-Teller and first-forbidden transitions. The
experimental total $\beta$-decay half-lives for the Ni isotopes with $A\leq$76
are described satisfactorily. The half-lives predicted from $A$=70 up to $A$=86
reveal fairly regular $A$-behaviour which results from simultaneous account for
the Gamow-Teller and first-forbidden transitions. For $Z\approx$ 28 nuclei, a
suppression of the delayed neutron emission probability is found when the
$N$=50 neutron closed shell is crossed. The effect originates from the
high-energy first-forbidden transitions to the states outside the $Q_{\beta} -
S_n$-window in the daughter nuclei.
  PACS numbers: 23.40.Bw,21.60.Jz,25.30.Pt,26.30.+k
