  We present the existence of the group inverse and its representation for the
block operator matrix $\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$
under the condition $FEF^{\pi}=0$. The group inverse for the anti-triangular
block matrices with two identical subblocks under the same condition is thereby
investigated. These extend the results of Zou, Chen and Mosi\'c (Studia Scient.
Math. Hungar., 54(2017), 489--508), and Cao, Zhang and Ge (J. Appl. Math.
Comput., 46(2014), 169--179).
