Ms Bishop has been deputy Liberal leader for over a decade but failed to win the majority vote in last Friday's leadership ballot, receiving only 11 votes.
The environment and energy portfolio, which was held by the man who is now Treasurer Josh Frydenberg, was also broken up in an acknowledgement of the tensions the two issues caused within the party.
The appointments come after Curtin MP Julie Bishop resigned as Foreign Minister to go to the back bench, one of only five women in the former Prime Minister Malcolm Turnbull's 25-minister Cabinet.
Steve Ciobo and Chirstopher Pyne are both being weighed up as a possible candidates for Ms Bishop's foreign affairs portfolio.
Abbott told 2GB on Monday he would "love to do a fair dinkum job" but wants to see "what this new role entails" because he did not want to "trip over the toes" of the Indigenous affairs minister.
Bishop's counterpart in the opposition Labor Party, Penny Wong, praised her "tireless work ethic", adding that her "commitment to standing up for Australia both here and overseas has never been in question".
Tyson Fury Quotes Frank Sinatra Then Gives Deontay Wilder Prediction
His team had been negotiating to fight WBA, WBO and IBF champion Anthony Joshua but talks fell through. This is what we've been waiting for here - the best fighting the best".
Bishop did not say whether she planned to stand in the next federal election in May 2019.
Australia's new prime minister announced a peace-making Cabinet on Sunday that does not punish his rivals in a bruising power struggle that ousted his predecessor days ago and divided a government that lags in opinion polls.
New prime minister Scott Morrison, who won the leadership ballot against Peter Dutton, is weighing up his new ministry to reunite the Liberal party. "She has set the bar for the next Foreign Minister very high".
Turnbull supporter David Coleman will take on Immigration, Citizenship and Multicultural Affairs in a clear signal from Morrison to separate the economic role of immigration from national security following policy disagreements with Dutton over the past year.
Senator Mathias Cormann, one of several senior ministers who had pledged their loyalty to Mr Turnbull before turning on him, will return as finance minister in Mr Morrison's new Cabinet.
And now it seems, the Liberal Party has lost her for good.
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