Hello and welcome
Looking at Frederick Cole's possible sister Elizabeth - in 1851 Elizabeth Gillmore b 1846 is living as granddaughter at St Helier with aunt Catherine Blake
John Burns 60 b Ireland
Elizabeth Burns 50 b Ireland
Eleanor Burns 11 daughter b England
Catherine Blake 21 married daughter b England
William Blake 4 Grandson b England
Ellen Blake 4 Months Grandaughter b Jersery St Helier
Elizabeth Gillmore 6 Granddaughter St Helier
A family tree on Ancestry lists John and Elizabeth Burns as having a daughter Elizabeth Burns born 1826 in Devon and includes a marriage cert for this Elizabeth Burns as a minor (father John Burns seaman) to William Gillmore of full age a Seaman on 19 May 1844 at Chapel of St Mary (Catholic) East Stonehouse Devon and with a daughter Elizabeth Gillmore born in 1845
Kay
Edit - Possibly Elizabeth Gillmore nee Burns in 1861 listed as a Mariners Wife and Tailoress which matches the occupation of Elizabeth Cole in 1871. They are living at St Helier (Could Edward Gilmore have become Gilmore Cole in 1871??)
Elizabeth Gilmore 30 b England
William Gilmore 9 b St Helier Jersey
Mary Gilmore 7 b St Helier Jersey
Edward Gilmore 3 St Helier Jersey
Hannah Whitley 24 Sister Widow
Henry Whitley 5 Nephew