Complex sentence: Some time after that they wrote a manifesto expressing their aim to oust the existing EDL from the top of Google's and Facebook's search results, aiming to achieve this via search engine optimisation using a Twitter feed and Facebook page.
Split into simpler sentences: Some time after that they wrote a manifesto expressing their aim to oust the existing EDL from the top of Google's and Facebook's search results. They aim to achieve this via search engine optimisation using a Twitter feed and Facebook page.