I looked up the birth of one of the children from the 1881 census on FreeBMD:
Births Dec quarter 1866
Surname First name(s) District Vol Page
Birch Peter Wirral 8a 359

When I put his mother's surname in as "Cole" it came up with not found.

There were two possible Joseph Birch reg. Wirral, neither with mother's surname Cole.

Births Sep 1878
Surname First name(s) District Vol Page
Birch Henry Wirral 8a 458

You need to get one of the children's birth certs to find out Margaret's surname, to be certain of this.

Sue