I tried the link I gave you myself and had problems with it; unchecking the box to the right of McDonough (Mother's last names) and then hitting search solved it however.
Moving on though; I did contact both my cousin and Tommy Burke. Tommy says his family moved to the Island after 1915 although he's doing some research. My cousin however, did provide the information I'm coping in below.
But first.. Did I mention that there are a number of researchers on the FBpage "Inishbofin Descendants"? Join by invitation only.
I (Tony Tierney) have both Edmund and Edward Burke in my database and have seen records where the name is recorded as Edward and another as Edmund. I tend to call both Edward as they are the same person irregardless of how their name was recorded in the past and it makes it easier for me to deal with the one name.
It would be interesting to see how Bridget Burke's family tree looks now as there is no information on her or Mary, so it is very likely that they both emigrated to USA and it looks like it was before the 1901 census anyway.
Here are some of the facts as I have them:
Edward Burke (age 27), the son of Edward Burke of Middle Quarter married Mary McDonough (age 30) the daughter of Michael McDonough of Middle Quarter on 15 Dec 1865 in St Colmans. James Joyce & Mary Cunnane were witnesses Reference SEMR-1865g
1867 12-Jan Edward Burke born to Edward Burke & Mary McDonagh Middle Quarter SEBR-1867a
1867 12-Jan Edward Burke son of Edward Burke & Mary McDonagh Death cert Middle Quarter SEDR-1867a
1868 31-Jan Patrick Burke son of Edward Burke & Mary McDonagh Middle Quarter SEBR-1868a Father's name recorded as Edmund
1870 1-May Bridget Burke daughter of Edward Burke & Mary McDonagh baptism sponsors John McDonagh & Bridget Scuffle Middle Quarter Father's name recorded as Edmund Ref TNECR-B & SEBR-1870d
1872 7-Dec Margaret Burke daughter of Edward Burke & Mary McDonagh Middle Quarter baptism sponsors were John & Bridget McDonagh Father's name recorded as Edmund Ref TNECR-B
1875 5-Nov Richard Burke son of Edward Burke & Mary McDonagh baptism sponsors were Mr McDonagh & Bridget Scuffle Middle Quarter Father's name recorded as Edmund Ref TNECR-B & SEBR-1875e
1879 31-Jan Mary Burke daughter of Edward Burke & Mary McDonagh baptism sponsors Michael McDonagh & Mary Burke Middle Quarter Father's name recorded as Edmund Ref TNECR-B & SEBR-1878h
The parents and two of their sons were still on Inishbofin for the 1901 census
Middle Quarter 41 Edward Burke head 60
Middle Quarter 41 Mary Burke wife 50
Middle Quarter 41 Patrick Burke son 30
Middle Quarter 41 Richard Burke son 25
Only the two sons remained there at the 1911 census
Middle Quarter 29 Burke Pat 40 Farmer - single
Middle Quarter 29 Burke Richard 37 Labourer - brother - single
Margaret married Redmond O'Halloran in 1897
OK Peggy.. Hope that helps you. As Tony mentions above, it would be nice to know where your family has gotten to since they left Ireland.