The group, which calls itself "Anonymous" and has been derided by Scientologists as a "pathetic" collective of "computer geeks".
However, in the last few weeks it has scored a couple of big successes, first by carrying out a denial of service attack on the Church of Scientology's international website, causing it to crash, and a sustained campaign of "Google bombing" - manipulating the way the internet search engine works - to ensure that the Church of Scientology is returned as the first hit whenever anyone enters the search string "dangerous cult".
The decision of hackers to target the self-styled church is believed to have stemmed from YouTube's decision to remove a video from the site showing Tom Cruise hailing Scientology as "a blast".
Anonymous allege that Scientologists forced YouTube to delete the highly embarrassing footage.
It forms part of a wider trend of "direct action" against the Church of Scientology. Global protests are planned for this Sunday, to voice concerns about the church's supposed love for "speech-suppression tactics" and "frivolous" legal injunctions to prevent criticism or discussion of the religion.
Protesters are mobilising online on sites such as Facebook and YouTube.
A video posted by Anonymous about its anti-Scientology campaign has been viewed more than 90,000 times, and the group has its own "channel" on the video-sharing site.
According to a press release circulated by the protest group, Anonymous said that that group's goals include "bringing an end to the financial exploitation of Church members and protecting the right to free speech".
It goes on to say that this alleged clampdown on free speech was "most evident on the recent attacks on websites such as Digg and YouTube, where the church filtered anti-Scientology comments and replaced their content with the text: 'This comment is no longer available due to a copyright claim by Church of Scientology International'."
The group also claims to have stolen a number of "secret documents" from the Church of Scientology's database, and says that these files have been made available on peer-to-peer file-sharing sites.
According to website WikiNews, the Church of Scientology has reported the web attacks to authorities.
"Activities of Anonymous have been reported to the Authorities and actions are being taken. Their activities are illegal and we do not approve of them. At the same time, our main work is to improve the environment, make people more able and spiritually aware. ... yes, we are taking action," said one Church of Scientology representative contacted by WikiNews.