  We determine the maximum lifetime t_max of 52 FRII radio sources found in 26
central group galaxies from cross correlation of the Berlind SDSS group catalog
with the VLA FIRST survey. Mock catalogs of FRII sources were produced to match
the selection criteria of FIRST and the redshift distribution of our parent
sample, while an analytical model was used to calculate source sizes and
luminosities. The maximum lifetime of FRII sources was then determined via a
comparison of the observed and model projected length distributions. We
estimate the average FRII lifetime is 1.5x10^7 years and the duty cycle is
~8x10^8 years. Degeneracies between t_max and the model parameters: jet power
distribution, axial ratio, energy injection index, and ambient density
introduce at most a factor of two uncertainty in our lifetime estimate. In
addition, we calculate the radio active galactic nuclei (AGN) fraction in
central group galaxies as a function of several group and host galaxy
properties. The lifetime of radio sources recorded here is consistent with the
quasar lifetime, even though these FRIIs have substantially sub-Eddington
accretion. These results suggest a fiducial time frame for energy injection
from AGN in feedback models. If the morphology of a given extended radio source
is set by large-scale environment, while the lifetime is determined by the
details of the accretion physics, this FRII lifetime is relevant for all
extended radio sources.
