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制作不容易，条件允许情况下，打赏一下，谢谢
作者使用xUtils下载文件，如使用其他框架下载，可在下载逻辑进行修改

AndroidManifest

   <!--使用网络-->
    <uses-permission android:name="android.permission.INTERNET" />
    <!-- 往SDCard 读写入数据权限 -->
    <uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE" />
    <!--安装apk权限-->
    <uses-permission android:name="android.permission.REQUEST_INSTALL_PACKAGES" />

<application

 android:usesCleartextTraffic="true"

...>

 <!--安装apk-->
        <!--非Androidx使用这个 android:name="android.support.v4.content.FileProvider"  -->
        <!-- Androidx使用这个 android:name="androidx.core.content.FileProvider"  -->
 <provider
            android:name="androidx.core.content.FileProvider"
            android:authorities="${applicationId}.fileProvider"
            android:exported="false"
            android:grantUriPermissions="true">
            <meta-data
                android:name="android.support.FILE_PROVIDER_PATHS"
                android:resource="@xml/file_paths" />
        </provider>

</application>

file_paths.xml

<?xml version="1.0" encoding="utf-8"?>
<paths>
   <external-path path="." name="external_storage_root" />
</paths>

下载逻辑和安装

public class MainActivity extends AppCompatActivity {

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        permission();
    }

    //点击下载按钮
    public void onDownload(View view) {
        if (permission() == false) {
            Toast.makeText(this, "未获取权限", Toast.LENGTH_SHORT).show();
            return;
        }
        downLoad();
    }

    //下载文件
    private void downLoad() {
        //apk下载链接
        String apkUrl = "http://xxx.apk";
        //文件路径
        String filePath = getExternalCacheDir() + "/Download/file.apk";
        //下载
        RequestParams params = new RequestParams(apkUrl);
        params.setSaveFilePath(filePath);//保存路径
        x.http().post(params, new Callback.ProgressCallback<File>() {
            @Override
            public void onSuccess(File result) {
                installAPK(MainActivity.this, result);
            }

            @Override
            public void onError(Throwable ex, boolean isOnCallback) {
            }

            @Override
            public void onCancelled(CancelledException cex) {
            }

            @Override
            public void onFinished() {
            }

            @Override
            public void onWaiting() {
            }

            @Override
            public void onStarted() {
            }

            @Override
            public void onLoading(long total, long current, boolean isDownloading) {
                Log.i("MainActivity", "正在下载：" + (int)((float)current/total * 100));
            }
        });
    }

    //安装apk
    public void installAPK(Context context, File apkFile) {
        Intent intent = new Intent(Intent.ACTION_VIEW);
        Uri uri;
        if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.N) {// 判断版本大于等于7.0
            uri = FileProvider.getUriForFile(context, context.getPackageName() + ".fileProvider", apkFile);
            intent.addFlags(Intent.FLAG_GRANT_READ_URI_PERMISSION | Intent.FLAG_ACTIVITY_NEW_TASK);
        } else {
            uri = Uri.parse("file://" + apkFile);
            intent.setFlags(Intent.FLAG_ACTIVITY_NEW_TASK);
        }
        intent.setDataAndType(uri, "application/vnd.android.package-archive");
        context.startActivity(intent);
    }

    //获取权限
    private boolean permission() {
        if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.M) {
            int REQUEST_CODE_CONTACT = 101;//授权回调标识
            //需要的权限
            String[] permissions = {
                    Manifest.permission.WRITE_EXTERNAL_STORAGE  //下载的apk保存到手机的权限
            };

            if (checkSelfPermission(permissions[0]) != PackageManager.PERMISSION_GRANTED) { //未授权
                requestPermissions(permissions, REQUEST_CODE_CONTACT);  //申请权限
                return false;
            } else {
                return true;
            }
        }
        return true;
    }

}