Segment Information - Financial Information Regarding Reportable Segments (Details) $ in Millions	12 Months Ended
	Jan. 01, 2023 USD ($) segment	Jan. 02, 2022 USD ($)	Dec. 27, 2020 USD ($)
Segment Reporting [Abstract]
Number of operating segments | segment	2
Number of reportable segments | segment	2
Segment Reporting Information [Line Items]
Revenues	$ 2,098.9	$ 1,837.4	$ 1,384.9
Operating expenses:
Selling, general and administrative	212.1	192.3	144.9
Depreciation and amortization	105.6	86.5	81.1
Total operating expenses	1,681.9	1,391.5	1,072.9
Operating profit	417.0	445.9	312.0
Interest expense, net	51.0	57.7	56.0
Other expense (income), net	4.6	0.2	(1.0)
Intercompany (income) expense	0.0	0.0	0.0
Income before income taxes	361.4	388.0	257.0
Income taxes	69.0	93.6	62.2
Net income	292.4	294.4	194.8
Total assets	3,362.9	3,154.3	2,528.9
Capital expenditures	178.3	135.6	69.8
Long-lived assets	1,587.7	1,362.5
U.S.
Operating expenses:
Long-lived assets	1,439.1	1,205.5
Foreign
Operating expenses:
Long-lived assets	148.6	157.0
Service revenues
Segment Reporting Information [Line Items]
Revenues	1,686.4	1,537.7	1,233.1
Operating expenses:
Cost of service and vehicle sales	996.5	851.5	721.7
Vehicle and parts sales
Segment Reporting Information [Line Items]
Revenues	412.5	299.7	151.8
Operating expenses:
Cost of service and vehicle sales	367.7	261.2	125.2
United States
Segment Reporting Information [Line Items]
Revenues	1,700.8	1,563.3	1,215.1
Operating expenses:
Selling, general and administrative	189.4	178.6	135.0
Depreciation and amortization	85.3	75.9	74.3
Total operating expenses	1,301.4	1,148.9	933.7
Operating profit	399.4	414.4	281.4
Interest expense, net	51.7	58.0	56.2
Other expense (income), net	5.8	0.5	(0.7)
Intercompany (income) expense	(10.3)	(8.5)	(8.0)
Income before income taxes	352.2	364.4	233.9
Income taxes	65.0	87.6	56.9
Net income	287.2	276.8	177.0
Total assets	2,775.8	2,510.1	2,341.1
Capital expenditures	166.3	124.9	52.3
United States | Service revenues
Segment Reporting Information [Line Items]
Revenues	1,539.7	1,429.2	1,134.4
Operating expenses:
Cost of service and vehicle sales	874.8	776.3	659.8
United States | Vehicle and parts sales
Segment Reporting Information [Line Items]
Revenues	161.1	134.1	80.7
Operating expenses:
Cost of service and vehicle sales	151.9	118.1	64.6
International
Segment Reporting Information [Line Items]
Revenues	398.1	274.1	169.8
Operating expenses:
Selling, general and administrative	22.7	13.7	9.9
Depreciation and amortization	20.3	10.6	6.8
Total operating expenses	380.5	242.6	139.2
Operating profit	17.6	31.5	30.6
Interest expense, net	(0.7)	(0.3)	(0.2)
Other expense (income), net	(1.2)	(0.3)	(0.3)
Intercompany (income) expense	10.3	8.5	8.0
Income before income taxes	9.2	23.6	23.1
Income taxes	4.0	6.0	5.3
Net income	5.2	17.6	17.8
Total assets	587.1	644.2	187.8
Capital expenditures	12.0	10.7	17.5
International | Service revenues
Segment Reporting Information [Line Items]
Revenues	146.7	108.5	98.7
Operating expenses:
Cost of service and vehicle sales	121.7	75.2	61.9
International | Vehicle and parts sales
Segment Reporting Information [Line Items]
Revenues	251.4	165.6	71.1
Operating expenses:
Cost of service and vehicle sales	$ 215.8	$ 143.1	$ 60.6