Calling Makefile in build.rs

I am trying to get a simple example of using build.rs to run Makefile commands but I have not had much success yet. Does anyone have an example of this working?
So far I have a small project with the following structure:
-makefile_integration
-----c_folder
---------c_test_lib.c (source file, library to build)
---------Makefile
-----src/main.rs
-----build.rs

Makefile is shown below. In c_folder, make and make clean create and destroy libc_test.a (along with other object files) in that folder as expected.

SRCS = c_test_lib.c
OBJS = $(SRCS:.c=.o)
TARGET_LIB = libc_test.a

CC = cc
RM = rm -f

$(TARGET_LIB): $(OBJS)
	ar rcs $@ $^

all: ${TARGET_LIB}

$(SRCS:.c=.d):%.d:%.c
	$(CC) -MM $< >$@

include $(SRCS:.c=.d)

clean:
	-${RM} ${TARGET_LIB} ${OBJS} ${SRCS:.c=.d}

.PHONY:		all clean

In build.rs, I have tried a couple of ways to get the same functionality from Cargo, but none are working.

let current = env::current_dir().unwrap();
 Command::new("make").env("PATH", current).args(&["-f", "c_folder/Makefile"]).status().unwrap();

This panics and cannot find the directory I want.

let c_folder = env::current_dir().unwrap().join("c_folder");
Command::new("make").current_dir(&Path::new(&c_folder)).status().unwrap();

This builds libc_test.a in c_folder (not somewhere in target) so main.rs does not have access to any C functions. And for whatever reason cargo clean does not clean this folder

Does anyone have an example of getting a build.rs to call a Makefile properly? Or know what I am doing wrong? I am trying to avoid using the Makefile to call cargo (though I know this is possible) and using cargo-make, and I think this should be possible with the proper methods on Command.

See this article:

https://kornel.ski/rust-sys-crate

In this case I would probably go for the "Replace the build system using cc" solution.