Hi Elaine
It might be simpler than you think. If this is the family in 1871 that you are referring to in 1881, there were family members by the name of TAYLOR/TAILOR. In this instance, son in law/dau in law can mean step son/daughter as well as the more usual in-law as we know it today. It looks as if Jane had previously been married to or had children in the name of TAYLOR. Further research should, hopefully, reveal whether she had a daughter named Eliza. If this is the case, then your Frederick would indeed have been a grandson of William RUMSEY albeit a step-grandson
1871
Prospect Cottage, Queens Rd, Millbrook, Hampshire
William RUMSEY head mar 54? tailor b.London
Jane wife 48 laundress b.Lymington
Henry TAILOR son in law 15/13 porter b.St Marys Southampton
Alice TAILOR dau in law 12 b.Shirley Southampton
Edward HARRIS son in law 31 seaman b.Shirley? Ports? Southampton
Emily HARRIS dau in law/wife 29 laundress b.Lymington
James LIGHT son in law 22 seaman b.Winchester
Mary Ann LIGHT dau in law/wife 22 b.Lymington
RG10/1200 folio 96 page 25
For future reference, it is often easier for anybody looking up if you post the full details of the 1881 census or whichever information you already hold.
Best wishes
Casalguidi