Set values of column based on other column

I have a data frame which looks like this:

  ID     Score New.ID New.Score
  123     5      456          
  456     1      789          
  789     0      123   

I would like to give the same scores to the New.ID column (which are just in a different order).

Desired result:

  ID    Score New.ID New.Score
  123     5      456         1
  456     1      789         0
  789     0      123         5

Code to reconstruct data frame:

ID <- as.factor(c(123,456,789))
Score <- c(5,1,0)
New.ID<- as.factor(c(456, 789, 123))
New.Score <- c(1,0,5)
dt <- data.frame(ID, Score, New.ID, New.Score)

Update

Desired output:

  Group  ID Score New.ID New.Score
     1 123     5    456         1
     1 456     1    789         0
     1 789     0    123         5
     2 555     1    999         0
     2 123     1    123         1
     2 999     0    555         1

So I am trying to attempt to use the function for each group. The ID 123 has score 5 in group 1, but in group 2 it has score 1. And I only want to use the scores that appear within each group.

I tried with ave:

mtch <- function(x) {
  dt[match(x,dt$ID),"Score"]  
}

dt$New.Score <- ave(dt$New.ID, dt$Group, FUN = mtch)

But it gives me NA values.

Code for second df:

Group <- as.factor(c(1, 1, 1, 2, 2, 2))
ID <- as.factor(c(123,456,789, 555, 123, 999))
Score <- c(5,1,0, 1,1,0)
dt <- data.frame(Group, ID, Score, New.ID)