I went to the website for the LDS www.familysearch.org

Had little success but he should be there in one form or another.

A not for profit History Foundation does have a birth for a
Henry Butt(?) in Kilkenny 1873, the only one before that is 1862 in Cork. There is a small fee.

The name Buttwell doesn't come up in the search just Butts and Botts but in 1869 is another name which might be
tied in but only a guess. Very few records for children with two names much less three and possibly that trend
might indicate a parent?

Oswald Hugh Ormiston Butt, Co. Down 1869.

Henry Hayes born 1866 and 1867 Limerick

Frederick born 1874, church baptism and civil registration for Kilkenny

Charlotte Hayes - all Ireland search 9 from 1849, which could explain no birth record for Charlotte as I would estimate her birth year at 1846-1847.
The dates even around the right time are 1849 Down and next is 1855 Cork.
There is a Charlotte Harriet, Offaly in 1868 which might of been a daughter who did not survive - again guessing.
Marriages for Charlotte Hayes close to 1867 are 1864 Co. Down and 1869 Kildare (too late-unless of course she didn't marry till Frederick was born)The different counties might mean different Army postings?

So now there are no births for either Henry or Frederick?
Would the British Army register them in England rather than Ireland?? I know there is a pay website that lists 'overseas' births of children for the Army. You could find this via the National Archives, Kew - military records

This particular resource serves most of Ireland's Counties, however, doesn't include Dublin.
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The 1871 UK Census show that Charlotte Buttwell was age 24 and lodging Queens Road, Aldershot, Surrey
son Frederick 2 and Henry 4, she is the wife of a soldier all born in Ireland.

This was the only result for the surname Buttwell in Ireland on a program that has a variety of records but which does not include births.

You could ask the question about British Army births in Ireland on a military message board, google MILGENIRE.
Military genealogy Ireland.

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