A lot depends on how long during the British tax year you've been back to blighty. Anything long that 183 days and you're considered a UK Tax payer (rules change if you constantly travel over 4 or more years). If you're only spending a few weeks out there then essentially you should not be expected to have any PAYE impact on your earnings. It also depends on how you are being paid - are you considered an employee or an independent contractor.

Don't forget, as a Green Card holder you are considered a permanent resident of the US and so are required to declare all your worldwide income to the IRS.

The US and UK do enjoy a treaty to help prevent double taxation but it doesn't always work the way you'd expect it to.