After dominating the first three Tests of the series, Australia are locked in a tense battle for a first-innings advantage.
Mitchell Johnson took 3-4 from four overs on Friday to finish with 5-63 in England's first innings of 255. The left-arm quick lifted his tally for the series to 28 victims at 14.96.
Rogers' third half-century of the series came off 133 balls and followed a nasty blow to the head after ducking into a bouncer from Stuart Broad with his score on 16.
The 36-year-old left-hander was treated on the field for a head wound and changed his helmet twice before settling into another gritty innings, which has included seven fours in three hours.
Australia had lost the wickets of David Warner (nine) and Shane Watson (10) before lunch.
Watson, batting with a groin strain, lunged forward and was caught behind off Ben Stokes after Warner top-edged a pull shot off Jimmy Anderson.
After taking seven wickets in the first three Tests, Anderson claimed a second victim for the day in a crucial blow to dismiss his arch rival Michael Clarke.
Clarke was clobbered over the back with a pad by Anderson in a dressing-room disagreement after the Adelaide Test seven summers ago.
Skipper Clarke played no shot on Thursday and looked on in horror as the ball nipped back off the seam to bowl him for 10.
New batsman Steve Smith was dropped by a diving Anderson at mid-wicket on seven.
Smith is 14 not out.
Australia are scoring at just 2.29 runs per over on a slow-paced MCG pitch which helped limit England's run rate to 2.55.
England had resumed on 6-226 and Johnston took three of the four wickets to fall.
Johnson grabbed 5-18 in nine overs with the second new ball after having 0-45 off his first 15 overs.
Star No.4 batsman Kevin Pietersen added four to his overnight score of 67.
Johnson struck twice in the second over of the day, removing Tim Bresnan caught at short leg and Pietersen bowled.
It was an awful slog shot from Pietersen, who had stood firm so brilliantly in the face of his critics on Thursday in four hours of determined batting.
ENGLAND FIRST INNINGS 255
AUSTRALIA FIRST INNINGS
C ROGERS not out 50 (135)
D WARNER c Bairstow b Anderson 9 (14)
S WATSON c Bairstow b Stokes 10 (15)
M CLARKE b Anderson 10 (34)
S SMITH not out 14 (54)
Sundries (3lb) 3
Three wickets for 96
Fall: 19 (Warner), 36 (Watson), 62 (Clarke).
Bowling: J Anderson 10-1-32-2, S Broad 8-2-19-0, B Stokes 9-4-17-1, T Bresnan 9-4-14-0, M Panesar 6-2-11-0.
Batting time: 180 mins. Overs: 42.
Umpires: Aleem Dar (PAK), Kumar Dharmasena (SRL).