You can create a new column where it's a combination of Group and Condition. For example, new column Class which will contains row as controlA, controlA, TreatedA, ...., controlC. Then it's easy to make dds object

#read the expression data in raw counts
readcounts <- read.table("expression_raw_counts.txt", header=TRUE, row.names = 1)

#read the sample information
sample_info <- read.table('sample_information.txt', sep='\t', header = TRUE, row.names = 1)

#Create a DESeq2 object named dds from the gene read count and sample information
dds <- DESeqDataSetFromMatrix(countData = readcounts,
                              colData = sample_info,
                              design = ~ Class)

#remove genes with 0 counts
keep_genes <- rowSums(counts(dds)) > 0
dds <- dds[ keep_genes, ]

#run deseq2
dds <- DESeq(dds, betaPrior=FALSE)

#generate results
res = results(dds, contrast = c("Class", "TreatedA", "controlA"))
write.table(res, "TreatedA_controlA.txt", sep='\t')

#run lfcShrink
res = lfcShrink(dds, contrast = c("Class", "TreatedA", "controlA"), res=res)
write.table(res, "TreatedA_controlA.lfc.txt", sep='\t')