Improving the character of school pupils was a central theme of former Education Secretary Nicky Morgan's policy agenda, with former children's minister Edward Timpson being given specific responsibility for "character and resilience" within the department.
Think-tank Demos has called for the DfE to refresh the character agenda within government, through a third round of character education grants, focused on developing good character online, after a survey it conducted found that 26 per cent of the 16- to 18-year-olds it questioned admitted to having "bullied or insulted someone else" online.
Of those who admitted to bullying somebody else, 93 per cent said that they had themselves experienced some form of cyber-bullying or abuse.
Demos said it found that young people's personal traits, values and skills may be significant in determining the extent to which they engage in positive or negative behaviours online.
In addition to boosting character education, Demos has called for the Department for Digital, Culture, Media and Sport to work with providers of the government's flagship National Citizen Service (NCS) to expand the digital component of the programme, so it promotes civic virtues and moral thinking online.
It also wants schools to deliver digital citizenship education that contains a strong emphasis on the moral implications of online social networking.
And it is keen for Facebook, and other social media providers, to work with youth charities and digital citizenship campaigns to develop effective ways of disseminating information that supports good character online.
Peter Harrison-Evans, researcher at Demos said: "Our findings show that online social networking can clearly facilitate risky or negative behaviours among a substantial minority of young people.
"Despite this, we caution against an overly restrictive response, not least because this can be counterproductive - encouraging more covert risky behaviour or limiting engagement in the positive aspects of social media, such as relationship building, and political and civic engagement.
"This research also shows the links between character traits such as empathy and self-control, and how young people think and act on social media.
"It's here that we feel policymakers, schools, and parents can make the biggest difference - empowering young people to make a positive contribution to their online communities by building their social digital skills and increasing their online moral sensitivity."