Not in the 1570's and 1580's. By then the Catholic faith had been overturned in England (despite the brief reign of Mary in the 1550's) for a generation. Those who remained Catholic throughout the 1540's, 50's, 60's and so on would have had to have been "staunch" to say the least.
It says, "family BACKGROUND."
Background IS one generation or more....and it was all catholic.
Technically, the Catholic faith, btw, was not overturned in England. The "Roman" variety of Catholicism was rejected.
The nation reverted to its Old English Catholic background that existed prior to William the Conqueror.