adjusted p value group wise or overall

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Recently I met an issue, when I do t.test like below:

library(dplyr)
library(rstatix)
data(ToothGrowth)
ToothGrowth%>%group_by(supp)%>%t_test(len~dose)
  supp  .y.   group1 group2    n1    n2 statistic    df            p      p.adj
* <fct> <chr> <chr>  <chr>  <int> <int>     <dbl> <dbl>        <dbl>      <dbl>
1 OJ    len   0.5    1         10    10     -5.05  17.7 0.0000878    0.000176  
2 OJ    len   0.5    2         10    10     -7.82  14.7 0.00000132   0.00000396
3 OJ    len   1      2         10    10     -2.25  15.8 0.039        0.039     
4 VC    len   0.5    1         10    10     -7.46  17.9 0.000000681  0.00000136
5 VC    len   0.5    2         10    10    -10.4   14.3 0.0000000468 0.00000014
6 VC    len   1      2         10    10     -5.47  13.6 0.0000916    0.0000916 

and you will notice that the adjusted p value were calculated within the "supp" group I can also calculate the adjusted p value with the p.adjust function

ToothGrowth%>%group_by(supp)%>%t_test(len~dose)%>%ungroup()%>%mutate(padj = p.adjust(p,method="holm"))%>%as.data.frame()
  supp .y. group1 group2 n1 n2  statistic       df        p    p.adj
1   OJ len    0.5      1 10 10  -5.048635 17.69835 8.78e-05 1.76e-04
2   OJ len    0.5      2 10 10  -7.817021 14.66780 1.32e-06 3.96e-06
3   OJ len      1      2 10 10  -2.247761 15.84238 3.90e-02 3.90e-02
4   VC len    0.5      1 10 10  -7.463430 17.86244 6.81e-07 1.36e-06
5   VC len    0.5      2 10 10 -10.387795 14.32712 4.68e-08 1.40e-07
6   VC len      1      2 10 10  -5.469814 13.59996 9.16e-05 9.16e-05
  p.adj.signif      padj
1          *** 2.634e-04
2         **** 5.280e-06
3            * 3.900e-02
4         **** 3.405e-06
5         **** 2.808e-07
6         **** 2.634e-04

So I am not sure which should be the correct way. I am thinking the first method is the correct one. But maybe someone will help me to answer the question and explain why? Thanks!

I also search some resources and try to have a clear idea but I may need to know more about this.

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