May Bettles from Northamptonshire

My grandmother's name was May Bettles. The information I have about her is not entirely consistent, and if you know anything about this family, please let me know: it is summarised in date order in this list:

 

Life Events
1939 register She is listed as May Jackson (her married name) at the address in Moray Road (London N4, where she lived right through to to 1955), as born on 30 March 1878.
1911 census

May Jackson at Moray Road, with my father's three elder siblings: he was the youngest, born during WW1. May's age is given as 33, which is consistent with the above. She says she was born in Wellingborough, Northamptonshire.

Marriage certificate Married on 10 Sept 1905, age given as 27, so born between 10/1877 and 10/1878, which is again consistent with the 1939 data. Her name is given as May, and her father's name is Alexander.
1901 census Mary Bettles is a servant at 34 Tollington Place (which is not far from Moray Road): she is aged 23 born Wellingborough: but note the subtle shift from May to Mary
1881 census piece RG11/1568

Mary S. Bettles, age 3 born Wellingborough, living at Hinwick, hamlet of Poddington (see next). Her parents are shown as Felix and Ann. (The immediately preceding entry is for John and Sophia Bettles, who may be her grandparents, but they are not part of the question.)
The entry also lists a yonger brother, Alexander Bettles, who was born about 1885 in Podington

Birth certificate Interesting. I have a birth certificate for a birth on 30 March 1878 in Wellingborough, an exact match with the 1939 register. However, the girl's name is Mary Sophia (Mary, not May, but it fits with the 1881 census), and her father's name is Frederick Bettles (Frederick, not Alexander as at marriage in 1905, or Felix as in the 1881 census): mother is Ann Maycock
Marriage certificate Felix (not Frederick as above) Bettles married Ann Maycock on 4 November 1877 in Bozeat Parish Church. (So Ann was pregnant with Mary at the time of marriage.)
   

So it looks as if she was christened Mary Sophia Bettles, in adulthood she did not use her second name, Sophia, at all, and for some reason took her first as May rather than Mary.
But was her father Frederick, Felix or Alexander?

   
1861 census RG9/952 Podington (entry 127 in census) lists John (age 28) and Sophia Bettles (25) with two sons Felix (age 8) and Alexander (2). Was one of these May/Mary's father?
Marriage certificate On 29 November 1850 John Bettle (no final -s) married Sophia Tye
   

If you think you can shed light on this family and the mixed up names, do let me know.