Whilst you can't technically appeal a dismissal, there is an opportunity to claim for either wrongful dismissal or mistaken identity if you can prove that the referee has made an obvious error. There are really strict deadlines to submit a claim as these need to be concluded within 7 days prior to the suspension commencing. In order to submit a claim you will need to log your intention to claim within 2 working days from the match, this can be done without the receiving the charge letter and there is no opportunity to extend this deadline. 

If you were sent off for anything other than for 2 cautions or for foul and abusive language, you have 2 working days from the date of the MATCH to submit an appeal through Whole Game, for example,  if you are sent off on a Sunday, you have until Tuesday night to confirm your intention to appeal. Anything over this time will not be considered and the dismissal will stand. You will then have a further 2 days to submit any evidence to Lancashire FA in support of the appeal if you have lodged your claim within the required period. The appeal will then be heard before the suspension begins, and the club will be informed of the result at the earliest opportunity.

If the claim is successful then the fee that you deposited will be returned. The red card would stand but the suspension and any fine would be rescinded.