I have quite a few Limmings/Lymings in Alton from 1545 until the early 1700s - could they be connected? I notice that there were only 4 of that name assessed for the 1665 Hampshire Hearth Tax - 2 in Alton, 1 in Odiham and 1 near Kingsclere - non in Crondall.
You mention that they were probably ‘bad boys’ to leave England in 1661. It is far more likely that they felt unable to stay in a country where the religion was changing from that of the Parliamentary period to that of Charles II. It was called ‘The Great Ejectment when almost 2000 dissenting congregations and their pastors were forced to leave the parish churches that had been their spiritual home for several years and meet where they could, in secrecy.’
A member of the Frost family of Binsted left England and ended up in Oyster Bay, Long Island, New York, at about the same time and it was not long after that that some local people became Quakers. People took their religion seriously then.
Yours Jane