Tax on state pension

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in Cutting tax
Just out of curiosity, am I to blame if the DWP fail to report my state pension to HMRC, and I am late paying income tax.
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My first payment was 6th April, but my company pension and income tax deduction for the end of April is already on my PTA online.
Clearly it is just hypothetical speculation, but when I spoke to HMRC in February, they said DWP would tell HMRC and I did not need to do anything.
DWP do notify HMRC when State Pension commences but HMRC won't issue a tax code with it included before it actually starts.
I presume DWP only notify HMRC around the time it does start and that an updated tax code will be issued shortly.
I guess they will need to split the state pension into 3 sections.
1) Up to and including 5th April 2022, at the old pension rate.
2) 6th April to 10th April 2022, at the old pension rate.
3) 11th April 2022 to 5th April 2023, at the new pension rate.
My company pension is paid monthly whereas my state pension is paid every four weeks.
The full new state pension is £185.15 per week which equates to £9,660.86 per year. (£185.15 / 7 x 365.25 = £9,660.86)
Will they calculate this as follows?
1) Reduce my personal allowance by £9,660.86
2) Reduce my tax code by 967
3) Increase the monthly tax deducted from my state pension by about £161 (£9,660 x 20% /12 = £161)
NB. From my the earlier discussion, this should be further adjusted as follows.
a) The increase in state pension only takes effect from 11th April 2022, so the annual state pension for 2022/2023 needs to be reduced. This will need an adjustment to include 5 days at the old rate of £179.16. I make this to be £3.96
b) A possible adjustment for unpaid tax in 2021/2022.
2. Your tax code allowances will be reduced by the State Pension amount to be coded in 2022:23
3. The extra tax will depend on how much your earnings are in relation to your tax code. If your earnings remain in the basic rate band then £161 would be about right.
a). Not sure HMRC work like that. I think it's much more likely to be 1 week at last year's rate and 51 at this year's rate.
b). Have you been notified of underpaid tax for 2021:22 as part of a 2021:22 tax code?