I have found more info
Written by kathleen (elder)
(2/16/2013 8:39 a.m.)
in consequence of the missive, maybe, penned by Stephanie
I didn't look at all of the notes in the Le Faye edition, so I missed the following:
Provenance.Given by FWA [Francis Austen] to Lord William FitzRoy, at the request of Mr William Upcott, who passed it on to Miss Catherine Hutton (1756-1846) for her autograph collection, 1841; ...
In his covering letter to Lord William FitzRoy FWA wrote:
The individual whose Autograph your friend is desirous to obtain was my Sister. I have several letters of hers in my possession, but not one that I could feel justified in parting with. I send you however her Signature such as she usually wrote it when she used (which she rarely did) more than her initials. The year is not affixed to the date, but it know it to have been 1813.
So the letter was probably written to either Frank or his wife.