Sum columns based on index in a a different data frame in R

I have two data frames similar to this:

df<-data.frame("A1"=c(1,2,3), "A2"=c(3,4,5), "A3"=c(6,7,8), "B1"=c(3,4,5))
ref_df<-data.frame("Name"=c("A1","A2","A3","B1"),code=c("Blue" ,"Blue","Green","Green"))

I would like to sum the values in the columns of df based on the code in the ref_df. I would like to store the results in a new data frame with column names matching the code in the ref_df

i.e. I would like a new data frame with Blue and Green as columns and the values representing the sum of A1+A2 and A3&B1 respectively. Like the one here:

result<-data.frame("Blue"=c(4,6,8), "Green"=c(9,11,13))

There are lots of post on summing columns based on conditions, but after a morning of research I cannot find any thing that solves my exact problem.