Research published by cyber security firm F-Secure suggested the attack was a "spear-phishing" campaign, in which people were sent targeted emails in attempts to fool them into clicking a rogue link or handing over their username and password, the BBC reported.
To do this, the attackers created a number of web addresses designed to resemble legitimate Foreign Office websites, including those used for accessing webmail.
F-secure said: "We are confident the Callisto Group used this type of access to a target's email account for the purposes of sending spear phishing to other targets. We also believe it is highly likely that the Callisto Group would leverage the same access to read and monitor the target's email activity.
"This targeting suggests the Callisto Group is interested in intelligence gathering related to foreign and security policy. Furthermore, we are unaware of any targeting in the described attacks that would suggest a financial motive."
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However, the company does not know whether the attack was successful.
The UK's National Cyber Security Centre is believed to have investigated the suspected attack, which began in April 2016, but has declined to say who was behind the attack on the Foreign Office.
"The government's Active Cyber Defence programme is developing services to block, prevent and neutralise attacks before they reach inboxes."
A cyber security expert at another company told the BBC on condition of anonymity that there was a link to the information uncovered in the investigation of Russian efforts to influence the US election.
F-Secure also told the BBC that it did notice some similarity between the Callisto Group's hacking and previous attacks that have been linked to Russia.
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