Mohammad Asyiq Tahir, 21, faces six charges, while his 27-year-old brother Mohammad Azhar Tahir faces 10 counts under the Computer Misuse and Cybersecurity Act.
The younger brother was accused of securing unauthorised access to a modem, unauthorised uploading of a video titled "Anonymous Retaliating and gives Warning" to someone else's YouTube account, and illegally accessing Gmail and Facebook accounts, among other things.
The elder brother allegedly modified the contents of the PMO website's server on November 7.
The alleged offences were said to have been committed over several days in early November, according to the Channel report.
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The first to face charges was 35-year-old Indian-origin man, James Raj Arokiasamy, who allegedly goes by the name "The Messiah".
Charged in mid-November, James was accused of compromising the website of Ang Mo Kio Town Council.
James Raj also faces four drug charges and is currently in remand to help with investigations.